Polynomial analogues of restricted multicolor b-ary partition functions

TL;DR

This paper introduces polynomial representations for restricted multicolor b-ary partition functions, providing explicit formulas, recurrence relations, and a factorization theorem to characterize these partitions.

Contribution

It develops a novel polynomial framework for restricted multicolor b-ary partitions, including explicit formulas and recurrence relations.

Findings

Polynomials characterize all restricted multicolor b-ary partitions.

Derived explicit formulas for the polynomials.

Established a recurrence relation and a factorization theorem.

Abstract

Given an integer base $b\geq 2$, a number $\rho\geq 1$ of colors, and a finite sequence $\Lambda=(\lambda_1,\ldots,\lambda_\rho)$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $\lambda_1+\cdots+\lambda_\rho$ variables, and prove that the $n$th polynomial characterizes all $\Lambda$-restricted $\rho$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.

Tables

Table 1. Table 1. Ω b , T Λ ​ ( n ; Z ) superscript subscript Ω 𝑏 𝑇 Λ 𝑛 𝑍 \Omega_{b,T}^{\Lambda}(n;Z) as in Example  2.5 , for 0 ≤ n ≤ 5 0 𝑛 5 0\leq n\leq 5 .

$n$	${\Omega }_{b,T}^{\Lambda }(n;Z)$
$0$	$1$
$1$	$z_1+y_1$
$2$	$z_1^{t_1}+z_2+z_1{y}_1+y_1^{s_1}+y_2$
$3$	$z_1^{1+t_1}+z_3+z_1^{t_1}{y}_1+z_2{y}_1+z_1{y}_1^{s_1}+y_1^{1+s_1}+z_1{y}_2$
$4$	$z_1^{t_1^2}+z_1^{t_1}{z}_2+z_2^{t_2}+z_1^{1+t_1}{y}_1+z_3{y}_1+z_1^{t_1}{y}_1^{s_1}$
	$+z_2{y}_1^{s_1}+y_1^{s_1^2}+z_1{y}_1^{1+s_1}+z_1^{t_1}{y}_2+z_2{y}_2+y_1^{s_1}{y}_2+y_2^{s_2}$
$5$	$y_1^{1+s_1^2}+y_1{y}_2^{s_2}+y_1^{s_1^2}{z}_1+y_1^{s_1}{y}_2{z}_1+y_2^{s_2}{z}_1+y_1^{1+s_1}{z}_1^{t_1}+y_1{z}_1^{t_1^2}+y_1^{s_1}{z}_1^{1+t_1}$
	$+y_2{z}_1^{1+t_1}+z_1^{1+t_1^2}+y_1^{1+s_1}{z}_2+y_1{z}_1^{t_1}{z}_2+y_1{z}_2^{t_2}+z_1{z}_2^{t_2}+y_1^{s_1}{z}_3+y_2{z}_3+z_1^{t_1}{z}_3$

Table 2. Table 2 . The monomials in ( 3.3 ).

Monomial	Partition	Rewritten
$z_{2,1}^{1+t_{2,1}}$	$1\cdot 2_2^0+1\cdot 2_2^1$	$2_2+1_2$
$z_{2,3}$	$3\cdot 2_2^0$	$1_2+1_2+1_2$
$z_{2,1}^{t_{2,1}}{z}_{1,1}$	$1\cdot 2_2^1+1\cdot 2_1^0$	$2_2+1_1$
$z_{2,2}{z}_{1,1}$	$2\cdot 2_2^0+1\cdot 2_1^0$	$1_2+1_2+1_1$
$z_{2,1}{z}_{1,1}^{t_{1,1}}$	$1\cdot 2_2^0+1\cdot 2_1^1$	$2_1+1_2$
$z_{1,1}^{1+t_{1,1}}$	$1\cdot 2_1^0+1\cdot 2_1^1$	$2_1+1_1$
$z_{2,1}{z}_{1,2}$	$1\cdot 2_2^0+2\cdot 2_1^0$	$1_2+1_1+1_1$

Table 3. Table 3. The terms of Ω 3 , T Λ ​ ( 6 ; Z ) superscript subscript Ω 3 𝑇 Λ 6 𝑍 \Omega_{3,T}^{\Lambda}(6;Z) and corresponding partitions.

Monomial	Partition	Monomial	Partition
$y_2^{s_2}$	$3_1+3_1$	$y_1{z}_1^{t_1}{z}_2$	$1_1+3_2+1_2+1_2$
$y_1^{s_1}{y}_2{z}_1$	$3_1+1_1+1_1+1_2$	$z_2^{t_2}$	$3_2+3_2$
$y_1^{s_1}{z}_1^{t_1}$	$3_1+3_2$	$y_1^{s_1}{z}_3$	$3_1+1_2+1_2+1_2$
$y_2{z}_1^{1+t_1}$	$1_1+1_1+1_2+3_2$	$z_1^{t_1}{z}_3$	$3_2+1_2+1_2+1_2$
$y_1^{1+s_1}{z}_2$	$1_1+3_1+1_2+1_2$

Table 4. Table 4. Ω 4 , T Λ ​ ( n ; Z ) superscript subscript Ω 4 𝑇 Λ 𝑛 𝑍 \Omega_{4,T}^{\Lambda}(n;Z) with Λ = ( 2 , 3 ) Λ 2 3 \Lambda=(2,3) , for 0 ≤ n ≤ 11 0 𝑛 11 0\leq n\leq 11 .

$n$	${\Omega }_{4,T}^{\Lambda }(n;Z)$	$n$	${\Omega }_{4,T}^{\Lambda }(n;Z)$
$0$	$1$	$6$	$\left(y_1^{s_1}+z_1^{t_1}\right)\left(y_2+y_1{z}_1+z_2\right)$
$1$	$y_1+z_1$	$7$	$\left(y_1^{s_1}+z_1^{t_1}\right)\left(y_2{z}_1+y_1{z}_2+z_3\right)$
$2$	$y_2+y_1{z}_1+z_2$	$8$	$y_2^{s_2}+\cdots +y_1{z}_1^{t_1}{z}_3$
$3$	$y_2{z}_1+y_1{z}_2+z_3$	$9$	$y_1{y}_2^{s_2}+\cdots +y_2{z}_1^{t_1}{z}_3$
$4$	$y_1^{s_1}+z_1^{t_1}+y_2{z}_2+y_1{z}_3$	$10$	$\left(y_2^{s_2}+y_1^{s_1}{z}_1^{t_1}+z_2^{t_2}\right)\left(y_2+y_1{z}_1+z_2\right)$
$5$	$y_1^{1+s_1}+\cdots +y_2{z}_3$	$11$	$\left(y_2^{s_2}+y_1^{s_1}{z}_1^{t_1}+z_2^{t_2}\right)\left(y_2{z}_1+y_1{z}_2+z_3\right)$

Equations

$$n=\sum _{j\ge 0}{c}_j{b}^j,c_j\in \{0,1,\dots ,\lambda \},$$

$${\omega }_{s,t}(2n;y,z)$$

$${\omega }_{s,t}(2n+1;y,z)$$

$${\omega }_{s,t}(n+1;y,z)=\sum _{h\in {\mathbb{H}}_n}{y}^{p_h(s)}{z}^{q_h(t)},$$

$$p_h(s)$$

$$q_h(t)$$

$$n=2^{{\alpha }_1}+\cdots +2^{{\alpha }_{\mu }}+(2^{{\beta }_1}+2^{{\beta }_1})+\cdots +(2^{{\beta }_{\nu }}+2^{{\beta }_{\nu }}).$$

$${\omega }_{s,t}(11;y,z)=z^{1+t^2}+y^s{z}^{t^2}+y^{s^2}{z}^{1+t}+y^{s^3}z+y^{s+s^3}.$$

$$\sum _{n=1}^{\infty }{\omega }_{s,t}(n;y,z)q^n=q\prod _{j=0}^{\infty }\left(1+y^{s^j}{q}^{2^j}+z^{t^j}{q}^{2^{j+1}}\right).$$

$${\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{\ast }\equiv \left(\genfrac{}{}{0ex}{}{n}{k}\right)(\mathrm{mod}2),{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{\ast }\in \{0,1\}.$$

$$d_t(n):=\sum _{j\ge 0}{c}_j{t}^j.$$

$${\omega }_{s,t}(n+1;y,z)=\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }{\left(\genfrac{}{}{0ex}{}{n-k}{k}\right)}^{\ast }{y}^{d_s(n-2k)}{z}^{d_t(k)}.$$

$${\omega }_{s,t}(n+1;y,z)=\sum _{k_1+2k_2=n}{\left(\genfrac{}{}{0ex}{}{k_1+k_2}{k_2}\right)}^{\ast }{y}^{d_s(k_1)}{z}^{d_t(k_2)}.$$

$$\omega (k\cdot 2^{\ell }+j;y,z)$$

$$\displaystyle\quad+\omega(k;y^{s^{\ell}},z^{t^{\ell}})\cdot\big{(}\omega(2^{\ell}+j;y,z)-y^{s^{\ell}}\omega(j;y,z)\big{)}.$$

$$\sum_{n=0}^{\infty}C_{b}^{\Lambda}(n)q^{n}=\prod_{j=0}^{\infty}\bigg{(}1+q^{b^{j}}+\cdots+q^{\lambda_{1}\cdot b^{j}}\bigg{)}\cdots\bigg{(}1+q^{b^{j}}+\cdots+q^{\lambda_{\rho}\cdot b^{j}}\bigg{)}.$$

$$\sum _{n=0}^{\infty }{C}_2^{\Lambda }(n)q^n$$

$$=1+2q+5q^2+7q^3+13q^4+17q^5+26q^6+\dots ,$$

$$\sum_{n=0}^{\infty}S_{b}^{\lambda}(n)q^{n}=\prod_{j=0}^{\infty}\bigg{(}1+q^{b^{j}}+q^{2\cdot b^{j}}+\cdots+q^{\lambda\cdot b^{j}}\bigg{)}.$$

$$\sum _{n=0}^{\infty }{\omega }_{b,T}^{\lambda }(n+1;Z)q^n=\prod _{j=0}^{\infty }\left(1+z_1^{t_1^j}{q}^{b^j}+z_2^{t_2^j}{q}^{2\cdot b^j}+\cdots +z_{\lambda }^{t_{\lambda }^j}{q}^{\lambda \cdot b^j}\right).$$

$${\omega }_{b,T}^{\lambda }(n+1;(1,\dots ,1))=S_b^{\lambda }(n).$$

$$\sum _{n=0}^{\infty }{\Omega }_{b,T}^{\Lambda }(n;Z)q^n=\prod _{j=0}^{\infty }$$

$$\left(1+z_{\rho ,1}^{t_{\rho ,1}^j}{q}^{b^j}+\cdots +z_{\rho ,{\lambda }_{\rho }}^{t_{\rho ,{\lambda }_{\rho }}^j}{q}^{{\lambda }_{\rho }\cdot b^j}\right).$$

$${\Omega }_{b,T}^{\Lambda }(n;(1,\dots ,1))=C_b^{\Lambda }(n),n=0,1,\dots .$$

$${\Omega }_{b,T}^{\Lambda }(n;Z)={\omega }_{b,T}^{\lambda }(n+1;Z),\rho =1,n=0,1,\dots .$$

$$\sum_{n=0}^{\infty}\Omega_{b,T}^{\Lambda}(n;Z)q^{n}=\prod_{j=0}^{\infty}\bigg{(}1+y_{1}^{s_{1}^{j}}q^{2^{j}}+y_{2}^{s_{2}^{j}}q^{2\cdot 2^{j}}\bigg{)}\bigg{(}1+z_{1}^{t_{1}^{j}}q^{2^{j}}+z_{2}^{t_{2}^{j}}q^{2\cdot 2^{j}}+z_{3}^{t_{3}^{j}}q^{3\cdot 2^{j}}\bigg{)}.$$

$$T_{\ell }=(t_{\ell ,1},\dots ,t_{\ell ,{\lambda }_{\ell }})and{Z}_{\ell }=(z_{\ell ,1},\dots ,z_{\ell ,{\lambda }_{\ell }}).$$

$${\Omega }_{b,T}^{\Lambda }(n;Z)=\sum _{n_1+\cdots +n_{\rho }=n}{\omega }_{b,T_1}^{{\lambda }_1}(n_1+1;Z_1)\cdots {\omega }_{b,T_{\rho }}^{{\lambda }_{\rho }}(n_{\rho }+1;Z_{\rho }).$$

$$2_2+1_2,2_2+1_1,2_1+1_2,2_1+1_1,1_2+1_2+1_2,1_2+1_2+1_1,1_2+1_1+1_1.$$

$${\Omega }_{2,T}^{\Lambda }(3;Z)=z_1^{1+t_1}+z_3+z_1^{t_1}{y}_1+z_2{y}_1+z_1{y}_1^{s_1}+y_1^{1+s_1}+z_1{y}_2.$$

$${\Omega }_{2,T}^{\Lambda }(3;Z)=z_{2,1}^{1+t_{2,1}}+z_{2,3}+z_{2,1}^{t_{2,1}}{z}_{1,1}+z_{2,2}{z}_{1,1}+z_{2,1}{z}_{1,1}^{t_{1,1}}+z_{1,1}^{1+t_{1,1}}+z_{2,1}{z}_{1,2}.$$

$${\Omega }_{b,T}^{\Lambda }(n;Z)=\sum _{h=1}^{C_b^{\Lambda }(n)}\prod _{\ell =1}^{\rho }{z}_{\ell ,1}^{p_{h,\ell ,1}(t_{\ell ,1})}\cdots z_{\ell ,{\lambda }_{\ell }}^{p_{h,\ell ,{\lambda }_{\ell }}(t_{\ell ,{\lambda }_{\ell }})},$$

$$p_{h,\ell ,i}(t_{\ell ,i})=t_{\ell ,i}^{{\tau }_{\ell ,i}(1)}+t_{\ell ,i}^{{\tau }_{\ell ,i}(2)}+\cdots +t_{\ell ,i}^{{\tau }_{\ell ,i}({\mu }_{\ell ,i})},{\mu }_{\ell ,i}\ge 0.$$

Full text

Research supported in part by the Natural Sciences and Engineering Research Council of Canada, Grant # 145628481

1. Introduction

The study of binary partitions and partition functions originated with Euler [10, p. 162ff.], but is still an active area of research. These concepts were later extended to $b$-ary and restricted $b$-ary partitions for arbitrary integer bases $b\geq 2$; some of the most important contributions were made, in chronological order, by Mahler [13], Churchhouse [3], Rødseth [15], Andrews [1], Reznick [14], and Dumont et al. [9]. Further details, putting these publication in perspective, can be found in [7].

More recently, Rødseth and Sellers [16] introduced and studied $b$-ary overpartitions, in analogy to ordinary overpartitions that had been introduced a little earlier by Corteel and Lovejoy [4]. Ordinary overpartitions were extended to restricted multicolor partitions by Keith [12], and colored $b$-ary partitions have recently been studied by Ulas and Żmija [18].

In almost all the papers mentioned above, the authors obtained interesting and important arithmetic properties of the partition functions in question, including Ramanujan-type congruences. The purpose of this paper is quite different. Building on earlier work, especially on our recent paper [7], we define a sequence of multivariable polynomials that characterize all individual partitions of an integer $n\geq 0$ (as opposed to just their numbers) for a large class of restricted colored $b$-ary partitions.

To make the results of the current paper more transparent and easier to understand, we recall some earlier work, which may serve as an example. Given integers $b\geq 2$ and $\lambda\geq 1$, the representations

[TABLE]

of an integer $n\geq 0$ are restricted $b$-ary partitions, and their numbers are denoted by $S_{b}^{\lambda}(n)$ in [7]. In particular, $S_{b}^{b-1}(n)=1$ since it corresponds to the unique $b$-ary representation of $n$. In the base 2 case, $S_{2}^{2}(n)$ counts the number of hyperbinary representations of $n$, and $S_{2}^{2}(n)=s(n+1)$, where $\{s(n)\}$ is the well-known Stern sequence; see [14, p. 470]. Related to this, Bates and Mansour [2] and independently Stanley and Wilf [17] introduced polynomial analogues as refinements of $s(n+1)$; their coefficients count hyperbinary representations with certain properties.

Recently the current authors [5] refined this further by introducing the following sequence of bivariate polynomials.

Definition 1.1** ([5], Definition 4.1).**

Let $s$ and $t$ be fixed positive integer parameters. We define the $2$-parameter generalized Stern polynomials in the variables $y$ and $z$ by $\omega_{s,t}(0;y,z)=0$, $\omega_{s,t}(1;y,z)=1$, and for $n\geq 1$ by

[TABLE]

By comparing the recurrence relations (1.2), (1.3) with those of Stern’s sequence, it is clear that $\omega_{s,t}(n;1,1)=s(n)$, independent of the parameters $s,t$. The usefulness of the polynomials $\omega_{s,t}(n;y,z)$, also relevant for the current paper, lies in the following characterization of all hyperbinary expansions of a positive integer $n$, which we denote by ${\mathbb{H}}_{n}$.

Theorem 1.2** ([5], Theorem 4.2).**

For any integer $n\geq 1$ we have

[TABLE]

where for each hyperbinary expansion $h$ of $n$, the exponents $p_{h}(s)$, $q_{h}(t)$ are polynomials in $s$ and $t$ respectively, with only [math] and $1$ as coefficients. Furthermore, if

[TABLE]

then the hyperbinary expansion $h\in{\mathbb{H}}_{n}$ is given by

[TABLE]

Example 1.3** ([5], Example 8).**

With Definition 1.1 we compute

[TABLE]

If we choose the third term on the right, then according to Theorem 1.2 the corresponding hyperbinary expansion will be $n=4+1+1+2+2$. The other four expansions can be “read off” analogously.

The sequence of polynomials $\omega_{s,t}(n;y,z)$ satisfies the following generating function.

Theorem 1.4** ([5], Theorem 4.4).**

For integers $s,t\geq 1$ we have

[TABLE]

Finally, there is an explicit formula, for which we first need to recall some notation. We consider binomial coefficients modulo 2 as follows:

[TABLE]

Furthermore, let $n=\sum_{j\geq 0}c_{j}2^{j}$, $c_{j}\in\{0,1\}$, be the binary expansion of an integer $n\geq 0$. Then for integers $t\geq 1$ we define

[TABLE]

We can now state a first version of the desired explicit formula.

Lemma 1.5** ([5], Theorem 4.3).**

For integers $s,t\geq 1$ and $n\geq 0$ we have

[TABLE]

As shown in [7], the identity (1.10) can be rewritten in a form that will extend more easily to a more general situation.

Theorem 1.6** ([7], Equation (55)).**

For integers $s,t\geq 1$ and $n\geq 0$ we have

[TABLE]

Finally, the polynomials $\omega_{s,t}(n;y,z)$ also satisfy various identities, an example of which will follow. For greater ease of notation we drop the subscripts $s,t$ and just write $\omega(n;y,z)$. We present a special case of Theorem 4.2 in [8].

Theorem 1.7**.**

Let $\ell\geq 1$ and $k\geq 1$ be integers. Then for all integers $j$ with $0\leq j\leq 2^{\ell}$ we have

[TABLE]

All this, from Definition 1.1 to Theorem 1.7, has been generalized, first to $b$-ary multivariable Stern polynomials in [8], and then further to polynomial analogues of restricted $b$-ary partition functions; see [7]. For general $b\geq 2$, some phenomena become visible that are trivial in the case $b=2$. For instance, the general case of (1.12) leads to a factorization result that is meaningful only for $b\geq 3$; see Corollary 4.3 in [8].

It is the purpose of this paper to obtain results for polynomial analogues of restricted colored $b$-ary partition functions, extending Theorems 1.2, 1.4, 1.6, and 1.7. We begin by defining, in Section 2, the concepts of restricted colored $b$-ary partitions, the corresponding partition functions, and their polynomial analogues. Section 3 contains the main result of this paper, namely a representation theorem that connects the restricted colored $b$-ary partitions of Definition 2.1 with the multivariate polynomials of Definition 2.4. In Section 4 we derive a recurrence relation for these polynomials, which we use in Section 5 to prove the representation theorem. In Sections 6 and 7 we obtain an explicit formula and a factorization result, respectively, for the multivariate polynomials; we then end with some further remarks and examples in Section 8.

2. Colored $b$-ary partition functions

The concepts of a restricted or unrestricted colored $b$-ary partition can be seen as an extension of $b$-ary overpartitions that were introduced by Rødseth and Sellers [16]. According to their definition, a $b$-ary overpartition of an integer $n\geq 1$ is a sequence of nonnegative integer powers whose sum is $n$, and where the first occurrence of a power $b$ may be overlined.

This means that each $b$-ary partition can have two types (or “colors”) of parts, namely the “basic” type which may occur an arbitrary number of times, and an “overlined” type, which may occur at most once. This leads to the idea of assigning a fixed number $\rho\geq 1$ of colors in a more symmetric fashion, that is, such that all colors are restricted.

Definition 2.1**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let $\Lambda=(\lambda_{1},\ldots,\lambda_{\rho})$ be a finite integer sequence with $1\leq\lambda_{1}\leq\cdots\leq\lambda_{\rho}$. Then a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\geq 1$ is a set of powers $b^{j}$ whose sum is $n$, and where for a given integer $j\geq 0$, up to $\lambda_{r}$ parts $b^{j}$ can be assigned the $r$th color, for $1\leq r\leq\rho$.

In other words, each part $b^{j}$ can be assigned the $\ell$th color at most $\lambda_{\ell}$ times, for $1\leq\ell\leq\rho$. If $C_{b}^{\Lambda}(n)$ denotes the number of partitions of $n$ as defined in Definition 2.1, then the generating function is

[TABLE]

We now illustrate Definition 2.1 with a concrete example.

Example 2.2**.**

Let $b=2$ and $\Lambda=(2,3)$, and thus $\rho=2$. Then the $\Lambda$-restricted 2-colored binary partitions of $n=4$ are**

$4_{2},\;4_{1},\;2_{2}+2_{2},\;2_{2}+2_{1},\;2_{1}+2_{1},\;2_{2}+1_{2}+1_{2},\;2_{2}+1_{2}+1_{1},\;2_{2}+1_{1}+1_{1},$**

$2_{1}+1_{2}+1_{2},\;2_{1}+1_{2}+1_{1},\;2_{1}+1_{1}+1_{1},\;1_{2}+1_{2}+1_{2}+1_{1},\;1_{2}+1_{2}+1_{1}+1_{1},$**

where the subscripts denote the colors of the part, and by convention we write each partition in non-increasing order of its parts, and assign the colors also in non-increasing order. This is consistent with the notation of colored ordinary partitions used by Keith [12]. Counting the partitions in this example, we see that $C_{2}^{\Lambda}(4)=13$. On the other hand, by (2.1) we have**

[TABLE]

which confirms that $C_{2}^{\Lambda}(4)=13$.**

When $\rho=1$, Definition 2.1 reduces to that of usual restricted $b$-ary partitions as in (1.1). Their numbers were denoted by $S_{b}^{\lambda}(n)$ in [7], with generating function

[TABLE]

When $\Lambda=(\lambda)$, then for all $n\geq 0$ we obviously have $C_{b}^{\Lambda}(n)=S_{b}^{\lambda}(n)$. The generating function (2.2) was then used in [7] as the basis for the following definition of a polynomial analogue.

Definition 2.3** ([7], Definition 1).**

Let $b\geq 2$ and $\lambda\geq 1$ be integers, $T=(t_{1},\ldots,t_{\lambda})$ be a $\lambda$-tuple of positive integer parameters, and $Z=(z_{1},\ldots,z_{\lambda})$ a $\lambda$-tuple of variables. Then we define the sequence of $\lambda$-variable polynomials $\omega_{b,T}^{\lambda}(n;Z)$ by the generating function

[TABLE]

We immediately see that this extends both (1.8) and (2.2). Indeed, if $T=(s,t)$, $Z=(y,z)$, and $b=\lambda=2$, then $\omega_{2,T}^{2}(n;Z)=\omega_{s,t}(n;y,z)$. Also, when $Z=(1,\ldots,1)$ and $T$ is arbitrary, then for any $b\geq 2$ and $\lambda\geq 1$ we have

[TABLE]

With (2.1) and (2.3) in mind, we are now ready to define the main object of study in this paper.

Definition 2.4**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let $\Lambda=(\lambda_{1},\ldots,\lambda_{\rho})$ be a finite integer sequence with $1\leq\lambda_{1}\leq\cdots\leq\lambda_{\rho}$. Furthermore, let $\lambda:=\lambda_{1}+\cdots+\lambda_{\rho}$ and let $T=(t_{1,1},\ldots,t_{1,\lambda_{1}};\ldots;t_{\rho,1},\ldots,t_{\rho,\lambda_{\rho}})$ be a $\lambda$-tuple of positive integer parameters, and $Z=(z_{1,1},\ldots,z_{1,\lambda_{1}};\ldots;z_{\rho,1},\ldots,z_{\rho,\lambda_{\rho}})$ a $\lambda$-tuple of variables. Then we define the sequence of $\lambda$-variable polynomials $\Omega_{b,T}^{\Lambda}(n;Z)$ by the generating function

[TABLE]

Suppose that in $Z$ we have $z_{i,j}=1$ for all relevant indices $i,j$. Then for arbitrary $T$ we have

[TABLE]

When $\rho=1$, we set $\Lambda=(\lambda)$ and $t_{1,i}=t_{i}$, $z_{1,i}=z_{i}$ for $1\leq i\leq\lambda$; then comparing (2.4) with (2.3) gives

[TABLE]

The argument $n+1$ on the right of this last identity is due to the fact that the polynomials $\omega_{b,T}^{\lambda}(n+1;Z)$ came from the study of generalized Stern polynomials; see also Definition 1.1.

We conclude this section by illustrating Definition 2.4 with an example that can be seen as a continuation of Example 2.2.

Example 2.5**.**

Let $b=2$ and $\Lambda=(2,3)$. To avoid double indices, we set $T=(s_{1},s_{2};t_{1},t_{2},t_{3})$ and $Z=(y_{1},y_{2};z_{1},z_{2},z_{3})$. Then the generating function (2.4) becomes**

[TABLE]

The first few polynomials, obtained by expanding the right-hand side of (2.7), are shown in Table 1. Note that the number of terms of the polynomials in Table 1 are 1, 2, 5, 7, 13, and 17, respectively. This is consistent with (2.5) and Example 2.2.**

We conclude this section with a result that is easily obtained by combining Definitions 2.3 and 2.4; it will be useful later, in Section 6.

Theorem 2.6**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let $\Lambda$, $T$, and $Z$ be as in Definition 2.4. Furthermore, for each $\ell=1,\ldots,\rho$ let

[TABLE]

Then for each integer $n\geq 0$ we have

[TABLE]

Proof.

We use the fact that the right-hand side of (2.4) is the product of $\rho\geq 1$ generating functions of the type that occurs on the right-hand side of the generating function (2.3). Equating coefficients of $q^{n}$, we see that the higher-order convolution on the right-hand side of (2.8) comes from the product of $\rho$ appropriate power series from the left of (2.3), while the left-hand side of (2.8) comes from the left of the identity (2.4). ∎

We note that in the special case $\rho=1$, the identity (2.8) reduces to (2.6).

3. A representation theorem

The main reason for introducing polynomial analogues for the Stern and $b$-ary Stern sequences [5, 6] is the fact that they can be used to characterize all hyperbinary and hyper $b$-ary representations. Theorem 1.2 and Example 1.3 may serve as an illustration of this. More generally, in [7] we used polynomial analogues for restricted $b$-ary partition functions to characterize all restricted $b$-ary partitions of a given integer.

In the present section we will extend these representation theorems to restricted colored $b$-ary partitions. We begin with an example.

Example 3.1**.**

As in Example 2.2 we let $b=2$ and $\Lambda=(2,3)$. This time we consider the $\Lambda$-restricted 2-colored binary partitions of $n=3$, namely**

[TABLE]

On the other hand, from Table 1 we have**

[TABLE]

We rewrite this polynomial in the notation of Definition 2.4, namely with $T=(t_{1,1},t_{1,2};t_{2,1},t_{2,2},t_{2,3})$ and $Z=(z_{1,1},z_{1,2};z_{2,1},z_{2,2},z_{2,3})$. Then (3.2) becomes**

[TABLE]

We assign each of the $C_{2}^{\Lambda}(3)=7$ partitions in (3.1) to one of the seven monomials in (3.3) as shown in Table 2.**

The one-to-one correspondence shown in Table 2 is a special case of the following representation theorem. Recall that $C_{b}^{\Lambda}(n)$ is the number of $\Lambda$-restricted colored $b$-ary partitions of $n$; see Definition 2.1.

Theorem 3.2**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let the finite sequences $\Lambda$, $T$, and $Z$ be as in Definition 2.4.

$(a)$* For any integer $n\geq 1$ we have*

[TABLE]

where for each $h=1,\ldots,C_{b}^{\Lambda}(n)$, the exponents $p_{h,\ell,i}(t_{\ell,i})$, $1\leq\ell\leq\rho$ and $1\leq i\leq\lambda_{\ell}$, are polynomials with only [math] and $1$ as coefficients.

$(b)$* We fix an $h=1,\ldots,C_{b}^{\Lambda}(n)$, and for each pair $(\ell,i)$ with $1\leq\ell\leq\rho$ and $1\leq i\leq\lambda_{\ell}$ we write*

[TABLE]

Let the exponents $\tau_{\ell,i}(j)$, $j=1,2,\ldots,$ form a strictly increasing finite sequence of nonnegative integers, and by convention we let $\mu_{\ell,i}=0$ correspond to $p_{h,\ell,i}(t_{\ell,i})=0$. Then the $\Lambda$-restricted colored $b$-ary partition of $n$ corresponding to the index-$h$* monomial on the right-hand side of (3.4) is*

[TABLE]

*where the subscript $\ell$ in $b_{\ell}$ indicates the *$\ell$th color of the part in question.

When $\rho=1$, Theorem 3.2 becomes a special case of Theorem 4 in [7]. By further restricting our attention to $\lambda_{1}=\lambda=b$, we obtain Theorem 11 in [6]; the case $b=2$ of this is presented as Theorem 1.2 above.

For the proof of Theorem 3.2 we require recurrence relations, which will be derived in the next section. We conclude the present section with an example.

Example 3.3**.**

Once again we take $b=2$ and $\Lambda=(2,3)$. From among the 17 terms of the entry for $\Omega_{2,T}^{\Lambda}(5;Z)$ in Table 1 we choose two monomials and rewrite them in terms of $T=(t_{1,1},t_{1,2};t_{2,1},t_{2,2},t_{2,3})$ and $Z=(z_{1,1},z_{1,2};z_{2,1},z_{2,2},z_{2,3})$, as in Example 3.1.

First we consider

[TABLE]

Using (3.5) and (3.6), we see that this monomial corresponds to the colored binary partition $1\cdot 2_{1}^{0}+1\cdot 2_{1}^{2}$, which we rewrite as $4_{1}+1_{1}$.

Next we consider the somewhat more interesting monomial

[TABLE]

Using (3.5) and (3.6) again, we get the partition $1\cdot 2_{1}^{0}+1\cdot 2_{2}^{1}+2\cdot 2_{2}^{0}$, which we rewrite as $2_{2}+1_{2}+1_{2}+1_{1}$.

Finally, we note that the correspondence between (3.5) and (3.6) also leads to the entries of Table 2.**

4. Recurrence relations

In this section we generalize the recurrence relations given by (1.2) and (1.3), and show that the polynomials $\Omega_{b,T}^{\Lambda}(n;Z)$ satisfy similar, though vastly extended recurrence relations. In the case of polynomial analogues of ordinary restricted $b$-ary partition functions, i.e., when $\rho=1$, we obtained such recurrence relations in [7]. Rewriting the relevant theorem in terms of the polynomials $\Omega_{b,T}^{\Lambda}(n;Z)$ by using the identity (2.6), we have the following result.

Theorem 4.1** ([7], Theorem 5).**

Let $b\geq 2$ and $\lambda\geq 1$ be integers, and set $\Lambda=(\lambda)$, so that $\rho=1$. Furthermore, let

[TABLE]

Then $\Omega_{b,T}^{\Lambda}(0;Z)=1$, and for integers $n\geq 0$ we have

[TABLE]

with the conventions that $z_{0}=1$, $z_{\mu}=0$ for $\mu>\lambda$, and $\Omega_{b,T}^{\Lambda}(m;Z)=0$ if $m<0$.

Example 4.2**.**

Let $\lambda=b=2$. Then, writing out the cases $j=0$ and $j=1$ separately, the identity (4.1) reduces to

[TABLE]

In this case, i.e., when $\Lambda=(2)$, we have $\Omega_{2,T}^{\Lambda}(n;z_{1},z_{2})=\omega_{t_{1},t_{2}}(n+1;z_{1},z_{2})$, and also $z_{0}=1$ and $z_{3}=0$. Therefore (4.2) and (4.3) are identical with (1.3) and (1.2), respectively.**

We now extend Theorem 4.1 to polynomial analogues of $\Lambda$-restricted colored $b$-ary partition functions.

Theorem 4.3**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let the finite sequences $\Lambda$, $T$, and $Z$ be as in Definition 2.4. Then $\Omega_{b,T}^{\Lambda}(0;Z)=1$, and for integers $n\geq 0$ we have

[TABLE]

with the convention that $\Omega_{b,T}^{\Lambda}(m;Z)=0$ if $m<0$. The coefficients $Y_{\nu}$, $0\leq\nu\leq\lambda$, are given by

[TABLE]

with the sum taken over all $i_{1},i_{2},\ldots,i_{\rho}$ with $i_{1}+\cdots+i_{\rho}=\nu$ and $0\leq i_{\ell}\leq\lambda_{\ell}$ for $\ell=1,\ldots,\rho$. In particular, $Y_{0}=1$, $Y_{1}=z_{1,1}+z_{2,1}+\cdots+z_{\rho,1}$, $Y_{\lambda}=z_{1,\lambda_{1}}z_{2,\lambda_{2}}\cdots z_{\rho,\lambda_{\rho}}$, and $Y_{\mu}=0$ when $\mu>\lambda$.

When $\rho=1$, we see that $Y_{\nu}=z_{1,\nu}=z_{\nu}$ for $0\leq\nu\leq\lambda$, and so Theorem 4.3 reduces to Theorem 4.1. Before proving Theorem 4.3, we present an example.

Example 4.4**.**

Continuing with Examples 2.5 and 3.1, we let $b=2$ and $\Lambda=(2,3)$, with $Z=(y_{1},y_{2};z_{1},z_{2},z_{3})$. For the sake of simplicity we delete the subscripts and superscripts of $\Omega$; Theorem 4.3 then gives

[TABLE]

where $Y_{0}=1$, $Y_{1}=y_{1}+z_{1}$, $Y_{2}=z_{2}+y_{1}z_{1}+y_{2}$, $Y_{3}=z_{3}+y_{1}z_{2}+y_{2}z_{1}$, $Y_{4}=y_{1}z_{3}+y_{2}z_{2}$, and $Y_{5}=y_{2}z_{3}$. As a specific example we take (4.7) with $n=2$. Then

[TABLE]

where we have used the first three entries in Table 1. Expanding the right-hand side of this last identity, we obtain the entry for $n=5$ in Table 1.**

Proof of Theorem 4.3.

For ease of notation we delete subscripts and superscripts from $\Omega$. By manipulating the infinite product in (2.4), we get

[TABLE]

We then obtain, again with (2.4),

[TABLE]

With the conventions $Y_{0}=1$ and $Y_{\mu}=0$ for $\mu>\lambda$, we get

[TABLE]

Next we write $m=bn+j$, $0\leq j\leq b-1$, in (4.8) and substitute (4.9) into (4.8). Then (4.8) becomes

[TABLE]

where we have used a Cauchy product, with $k+\nu=n$. Finally, by equating coefficients of $q^{nb+j}$ in the first and the third line of (4.10), we immediately get the desired identity (4.4). ∎

We finish this section by explicitly stating a functional equation that was derived as part of the proof of Theorem 4.3. We set

[TABLE]

then the identity (4.8) can be rewritten as follows.

Corollary 4.5**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, let the finite sequences $\Lambda$, $T$, and $Z$ be as in Definition 2.4, and the polynomials $Y_{\nu}$, $1\leq\nu\leq\lambda$, as in (4.5). Then the generating function $F(Z,q)$, defined by (4.11), satisfies the functional equation

[TABLE]

5. Proof of Theorem 3.2

In this section we use the recurrence relation, Theorem 4.3, to prove our representation result, namely Theorem 3.2. In doing so, we follow the outline of the proof of Theorem 4 in [7], although the details will be different. In order to simplify terminology, if there is no danger of ambiguity we will refer to a $\Lambda$-restricted $\rho$-colored $b$-ary partition, as defined in Definition 2.1, simply as “a partition of $n$”.

We fix the integers $b\geq 2$ and $\rho\geq 1$, as well as the $\rho$-tuple $\Lambda$, integer $\lambda$, and the $\lambda$-tuples $T$ and $Z$ as in Definition 2.4. As we did in the proof of Theorem 4.3, we suppress the subscripts and superscripts of $\Omega$. We proceed by induction on $n$.

1. For the induction beginning we consider (4.4) with $n=0$. Then

[TABLE]

where by (4.5) we have

[TABLE]

On the other hand, the only partitions of $j$, with $0\leq j\leq b-1$, are

[TABLE]

This is consistent with (3.4)–(3.6). Indeed, in view of (5.2), the identity (3.4) becomes

[TABLE]

This means that, in view of (3.5), for each $\ell=1,\ldots,\rho$ we have

[TABLE]

and so by (3.6) the corresponding partition of $j$ is

[TABLE]

which is consistent with (5.3). This completes the induction beginning.

2. Next we assume that Theorem 3.2 is true for all $m\leq nb-1$, for some $n\geq 1$. Our goal is to show that it is also true for $m=nb+j$, for all $j=0,1,\ldots,b-1$. To do so, we first fix an arbitrary $j$ with $0\leq j\leq b-1$. We then consider all partitions of $nb+j$, which can be obtained recursively as follows. We fix an integer $k\geq 0$ and

1. (a)

Take all partitions of $n-k$ and multiply each power of $b_{\ell}$ (including the 0th power) by $b_{\ell}$, $\ell=1,2,\ldots,\rho$, thus raising all powers by 1. 2. (b)

To each of the“raised” partitions from (a), add all partitions of $kb+j$ of the form

[TABLE] 3. (c)

Do (a) and (b) for all $k\geq 0$ that satisfy $kb+j\leq\lambda$.

This procedure gives all partitions of $nb+j$ since $(n-k)\cdot b+(kb+j)=nb+j$. Also, the maximal $k$ given by (c) is $\lfloor\lambda/b\rfloor$.

3. Using the induction hypothesis and (3.4), we have

[TABLE]

with exponents $p_{h,\ell,1}(t_{\ell,1}),\ldots,p_{h,\ell,\lambda_{\ell}}(t_{\ell,\lambda_{\ell}})$ as in (3.5). In order to lift the partitions of $n-k$ to those of $(n-k)b$, which corresponds to step (a), all powers of $t_{\ell,i}$, $1\leq\ell\leq\rho$ and $1\leq i\leq\lambda_{\ell}$, are augmented by 1. This, in turn, means that (5.5) changes to $\Omega(n-k;Z^{T})$.

Next, for each $k$ with $0\leq k\leq\lfloor\lambda/b\rfloor$, we multiply $\Omega(n-k;Z^{T})$ by the monomial

[TABLE]

which corresponds to an individual partition of $kb+j$ of the form (5.4) in step (b). This means that for each $\ell=1,\ldots,\rho$, the exponents of $z_{\ell,i_{\ell}}$ are increased by 1, which is again consistent with the duality between monomials and individual partitions. Now, taking into account all partitions of $kb+j$ of the form (5.4) means multiplying $\Omega(n-k;Z^{T})$ by

[TABLE]

But, by (4.5), this sum is exactly $Y_{kb+j}$.

4. Recall that steps 2 and 3 were done for an arbitrary fixed integer $k$ with $0\leq k\leq\lfloor\lambda/b\rfloor$. The polynomial corresponding to all partitions of $nb+j$ is therefore

[TABLE]

which, by Theorem 4.3, is $\Omega(bn+j;Z)$. Since $j$ was an arbitrarily chosen but fixed integer with $0\leq j\leq b-1$, this concludes the proof of Theorem 3.2 by induction.

6. Explicit formulas

In this section we derive explicit formulas for our polynomials $\Omega_{b,T}^{\Lambda}(n;Z)$, thus generalizing Theorem 1.6. For greater clarity, we begin with an intermediate result for the binary case, i.e., the case $b=2$. The function $d_{t}(n)$ is defined as in (1.9), and the “starred” multinomial coefficient is defined to be the least nonnegative residue modulo 2 of the original multinomial coefficient.

Theorem 6.1** ([7], Corollary 17).**

Let $\lambda\geq 1$ be an integer, and $Z$ and $T$ as in Definition 2.3. Then for $n\geq 0$ we have

[TABLE]

We see that for $\lambda=2$ the identity (6.1) reduces to (1.11). On the other hand, Theorem 6.1 was extended in [7] to arbitrary bases $b\geq 2$. Since this more general case will be needed to obtain the main result of this section, we state it here, but without proof. We need some additional definitions with corresponding notations; see also [7].

First we define the set

[TABLE]

i.e., the set of all nonnegative integers whose $b$-ary digits are only 0 or 1. Clearly we have $M_{2}={\mathbb{N}}\cup\{0\}$.

Next we extend (1.9) as follows. Let $k\in M_{b}$ with $k=\sum_{j\geq 0}c_{j}b^{j}$, $c_{j}\in\{0,1\}$. Then for an integer base $t\geq 1$ we define

[TABLE]

It is clear that $d_{t}^{2}(k)=d_{t}(k)$ and $d_{b}^{b}(k)=k$ for all integers $b\geq 2$ and $k\in M_{b}$.

Finally, we extend the multinomial coefficient modulo 2, as it is used in Theorem 6.1. If $k_{1},\ldots,k_{\lambda}\in M_{b}$, we set

[TABLE]

with the right-hand side of (6.4) as defined earlier. The desired generalization of Theorem 6.1 is then as follows.

Theorem 6.2** ([7], Theorem 21).**

Let $b\geq 2$ and $\lambda\geq 1$ be integers, and $T$ and $Z$ as in Definition 2.3. Then for $n\geq 0$ we have

[TABLE]

We note that Theorem 6.2 immediately implies Theorem 6.1. Indeed, when $b=2$, the multinomial coefficient defined in (6.4) reduces to the one in (6.1) since $d^{2}_{2}(k_{i})=k_{i}$. Then in the $b=2$ case we also have $d^{b}_{t_{i}}(k_{i})=d_{t_{i}}(k_{i})$, and the conditions concerning $M_{b}$ become irrelevant.

We are now ready to state and prove the main result of this section.

Theorem 6.3**.**

Let $b\geq 2$ and $\rho\geq 1$ be integers, and let $\Lambda$, $T$, and $Z$ be as in Definition 2.4. Furthermore, let

[TABLE]

Then for each integer $n\geq 0$ we have

[TABLE]

where the sum is taken over all integers $k_{\ell,i}\in M_{b}$ with $1\leq i\leq\lambda_{\ell}$ and $1\leq\ell\leq\rho$, such that

[TABLE]

Proof.

We combine Theorem 2.6 with Theorem 6.2, where in (6.1) we replace $n$ by $n_{\ell}$, $\lambda$ by $\lambda_{\ell}$ and add the additional subscript $\ell$ to $k_{i}$, $t_{i}$ and $z_{i}$, where $\ell=1,\ldots,\rho$. The right-hand side of (6.6) then follows immediately from the right-hand side of (2.8), and the condition (6.7) follows from the summation conditions in (2.8) and in (6.5). ∎

We conclude this section with two examples. The first one is our “running example”, continuing Examples 2.5, 3.1, and 4.4.

Example 6.4**.**

Let $b=2$ and $\Lambda=(2,3)$, and to avoid double indices, we set again $T=(s_{1},s_{2};t_{1},t_{2},t_{3})$ and $Z=(y_{1},y_{2};z_{1},z_{2},z_{3})$. We also set $h_{i}:=k_{1,i}$, $i=1,2$, and $k_{i}:=k_{2,i}$, $i=1,2,3$. Then by Theorem 6.3 we have

[TABLE]

Since $b=2$, the condition in Theorem 6.3 that concerns $M_{b}$ becomes irrelevant; see the remark following Theorem 6.2.

As a specific example we consider $n=4$. By direct computation we find that the solutions $(h_{1},h_{2};k_{1},k_{2},k_{3})$ of $h_{1}+2h_{2}+k_{1}+2k_{2}+3k_{3}=4$ for which the binomial/trinomial coefficients in (6.8) are odd (i.e., the “starred” versions are 1) are as follows:

[TABLE]

so that (6.8) becomes

[TABLE]

This is consistent with the entry for $\Omega_{2,T}^{\Lambda}(4;Z)$ in Table 1.

It is interesting to note that the smallest $n$ for which all three entries in the trinomial coefficient in (6.8) are nonzero is $n=11$, with $k_{1}=4$, $k_{2}=2$, and $k_{3}=1$, giving

[TABLE]

This, in turn, leads to the following monomial and corresponding colored binary partition:

[TABLE]

Example 6.5**.**

We now choose $b=3$, but leave $\Lambda$, $T$, and $Z$ as in Example 6.4. Then in analogy to (6.8) we get

[TABLE]

This time we consider $n=6$ as a specific example. The conditions $h_{1},\ldots,k_{3}\in M_{3}$ mean, in particular, that 2 and 5 cannot occur among the solutions of $h_{1}+2h_{2}+k_{1}+2k_{2}+3k_{3}=6$. By direct computation, using the definition (6.4), we find that both of the modified binomial/trinomial coefficients in (6.9) are 1 when $(h_{1},h_{2};k_{1},k_{2},k_{3})$ is of the form

[TABLE]

so that (6.9) becomes

[TABLE]

Finally we illustrate in Table 3 how the polynomial $\Omega_{3,T}^{\Lambda}(6;Z)$ translates to the set of nine $(2,3)$-restricted 2-colored ternary partitions of $n=6$. **

7. Some product identities

In dealing with polynomials in one or more variables with integer coefficients, questions of divisibility and irreducibility become relevant. For instance, in the case of multivariate $b$-ary Stern polynomials [8], we obtain an analogue of Theorem 1.7. We present a special case of this result, rewritten in the notation of the current paper. It corresponds to $\rho=1$ and $\Lambda=(\lambda)=(b)$.

Theorem 7.1** ([8], Corollary 4.3).**

Let $b\geq 2$, $\ell\geq 1$, and $n\geq 1$ be integers. Then

[TABLE]

In this section we show that Theorem 7.1 can be extended to an arbitrary number of colors $\rho$ and an arbitrary $\Lambda=(\lambda_{1},\ldots,\lambda_{\rho})$. We use again the notations of Definition 2.4.

Theorem 7.2**.**

Let $b\geq 2$, $\ell\geq 1$, and $n\geq 1$ be integers. Then

[TABLE]

which in the case $b\leq\lambda=\lambda_{1}+\cdots+\lambda_{\rho}$ holds for all integers $j$ with

[TABLE]

When $b>\lambda$, then (7.2) holds for all $j$ with $0\leq j\leq b^{\ell}-1$.

Proof.

We proceed by induction on $\ell\geq 1$, and as usual we suppress the subscripts and superscripts of $\Omega$. When $\ell=1$ and $\lambda-b+1\leq j\leq b-1$, then the recurrence relation (4.4) gives

[TABLE]

since $b+j\geq\lambda+1$, so all other coefficients $Y_{bk+j}$ in (4.4) vanish. For $n=0$, the identity (7.4) gives $\Omega(j;Z)=Y_{j}$, and we get

[TABLE]

In the case $\lambda<b$, (7.5) holds for all $0\leq j\leq b-1$. This establishes the induction beginning.

Now suppose that Theorem 7.2 holds for some $\ell\geq 1$. We wish to show that it also holds for $\ell+1$. To do so, we begin by setting

[TABLE]

This means that $j$ satisfies the inequalities

[TABLE]

as required. Now we have

[TABLE]

where we have used the induction hypothesis (7.2). Reordering this last line, we have

[TABLE]

where we have used (7.4) again. Note that in several steps the inequalities in (7.6) were essential. This completes the proof by induction. ∎

Example 7.3**.**

We choose $b=4$, $\Lambda=(2,3)$, and leave $T$ and $Z$ as in Example 6.4. Using the recurrence relation in Theorem 4.3 we can compute $\Omega_{4,T}^{\Lambda}(n;Z)$ for $0\leq n\leq 11$, and factor them when possible; see Table 4.

We note that the entries in Table 4 are consistent with Theorem 7.2. The entries for $n=5,8$, and 9 are irreducible and consist of 5, 7, and 8 monomials, respectively.**

As far as Theorem 1.7 is concerned, we believe that it extends in this form to general $\rho\geq 1$ and $\Lambda=(\lambda_{1},\ldots,\lambda_{\rho})$ only when $1\leq(\lambda+1)/b<2$, that is, when $b-1\leq\lambda_{1}+\cdots+\lambda_{\rho}\leq 2b-2$. This would be the case, for instance, in Example 7.3. Pursuing this question further would be beyond the scope of this paper.

8. Further Remarks

We recall that by setting $Z=(1,\ldots,1)$, we get the numerical sequences $C_{b}^{\Lambda}(n)=\Omega_{b,T}^{\Lambda}(n;(1,\ldots,1))$; see (2.5). While it is not the purpose of this paper to study these sequences, we present one easy but rather surprising result.

Corollary 8.1**.**

Let $b\geq 2$ be an integer, and let $\Lambda=(b-1,\ldots,b-1)$ with $\rho\geq 1$ elements. Then the number of $\Lambda$-restricted $\rho$-colored $b$-ary partitions is independent of $b$, namely

[TABLE]

Proof.

We use the identity

[TABLE]

which reflects the fact that each integer $n\geq 0$ has a unique $b$-ary expansion, or, in other words, $S_{b}^{b-1}(n)=1$. Now we raise the left- and the right-most terms in (8.2) to the power $\rho$ and use the well-known binomial expansion

[TABLE]

(see, e.g., [11, Eq. (1.3)]). Then the generating function (2.1) immediately gives the desired identity (8.1). ∎

Example 8.2**.**

Let $\rho=3$ and $n=3$. When $b=2$, we have $\Lambda=(1,1,1)$, and the $\binom{5}{2}=10$ allowable partitions are**

$2_{3}+1_{3},\;2_{3}+1_{2},\;2_{3}+1_{1},\;2_{2}+1_{3},\;2_{2}+1_{2},\;2_{2}+1_{1},\;2_{1}+1_{3},\;2_{1}+1_{2},\;2_{1}+1_{1},$**

$1_{3}+1_{2}+1_{1}.$**

For $b=3$, and thus $\Lambda=(2,2,2)$, we have**

$3_{3},\;3_{2},\;3_{1},\;1_{3}+1_{3}+1_{2},\;1_{3}+1_{3}+1_{1},\;1_{3}+1_{2}+1_{2},\;1_{3}+1_{2}+1_{1},$**

$1_{3}+1_{1}+1_{1},\;1_{2}+1_{2}+1_{1},\;1_{2}+1_{1}+1_{1},$**

and for any $b\geq 4$ we have the partitions**

$1_{3}+1_{3}+1_{3},\;1_{3}+1_{3}+1_{2},\;\ldots,\;1_{2}+1_{1}+1_{1},\;1_{1}+1_{1}+1_{1}.$**

This last example is an instance of the more general situation where $n<b$, in which case all partitions have parts 1 with $\rho$ different colors. These colors may be assigned $b-1\geq n$ times, which means that there are no restrictions. Therefore we have a one-to-one correspondence between such partitions and all non-increasing sequences of length $n$ consisting of integers $1,2,\ldots,\rho$. The number of such sequences is a known combinatorial quantity, namely $\binom{n+\rho-1}{\rho-1}$, consistent with Corollary 8.1.**

Finally we return to the main topic of this paper, namely the multivariate polynomials that characterize all $\Lambda$-restricted $\rho$-colored $b$-ary partitions of an integer $n\geq 1$.

Example 8.3**.**

As in Example 8.2 we let $\rho=3$ and $n=3$. Using any of the various methods described in this paper for computing the relevant polynomials, we obtain the following expressions.

(a) Let $b=2$ and $\Lambda=(1,1,1)$. Then with $Z=(x_{1};y_{1};z_{1})$ and $T=(r_{1};s_{1};t_{1})$ we get

[TABLE]

(b) Let $b=3$ and $\Lambda=(2,2,2)$. Then with $Z=(x_{1},x_{2};y_{1},y_{2};z_{1},z_{2})$ and $T=(r_{1},r_{2};s_{1},s_{2};t_{1},t_{2})$ we get

[TABLE]

(c) Let $b\geq 4$ and $\Lambda=(b-1,b-1,b-1)$. Then with $Z=(x_{1},\ldots,x_{b-1};y_{1},\ldots,y_{b-1};$ $z_{1},\ldots,z_{b-1})$ we get

[TABLE]

Here in (8.5) we did not specify $T$ because no exponents other than 1 occur.

All these polynomials (8.3)–(8.5) consist of ten monomials, which is consistent with Corollary 8.1 and Example 8.2. Each of these polynomials characterizes the three sets of partitions listed in Example 8.2, in the order in which the respective monomials are given.**