Characterizing the number of coloured $m$-ary partitions modulo $m$,   with and without gaps

I. P. Goulden; Pavel Shuldiner

arXiv:1701.07077·math.CO·January 26, 2017·Discret. Math.

Characterizing the number of coloured $m$-ary partitions modulo $m$, with and without gaps

I. P. Goulden, Pavel Shuldiner

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TL;DR

This paper extends previous work on the modulo properties of $m$-ary partitions to coloured $m$-ary partitions, providing explicit generating function expansions and new characterizations.

Contribution

It introduces a novel proof method and explicit generating function expansions for coloured $m$-ary partitions modulo $m$, with and without gaps.

Findings

01

Explicit expansions for generating functions modulo m

02

Complete characterization of coloured m-ary partitions modulo m

03

Extension of previous results to coloured partitions

Abstract

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of $m$ -ary partitions modulo $m$ , with and without gaps. In this paper we extend these results to the case of coloured $m$ -ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo $m$

Equations61

(1 - q)^{- a} \equiv (1 - q^{m})^{- 1} ℓ = 0 \sum m - 1 (a - 1 a - 1 + ℓ) q^{ℓ} (mod m) .

(1 - q)^{- a} \equiv (1 - q^{m})^{- 1} ℓ = 0 \sum m - 1 (a - 1 a - 1 + ℓ) q^{ℓ} (mod m) .

(1 - q)^{- a} = ℓ = 0 \sum \infty (a - 1 a - 1 + ℓ) q^{ℓ} .

(1 - q)^{- a} = ℓ = 0 \sum \infty (a - 1 a - 1 + ℓ) q^{ℓ} .

(a - 1 a - 1 + ℓ) = ((a - 1)!)^{- 1} (a - 1 + ℓ)_{a - 1} .

(a - 1 a - 1 + ℓ) = ((a - 1)!)^{- 1} (a - 1 + ℓ)_{a - 1} .

(a - 1 + ℓ + m)_{a - 1} \equiv (a - 1 + ℓ)_{a - 1} (mod m),

(a - 1 + ℓ + m)_{a - 1} \equiv (a - 1 + ℓ)_{a - 1} (mod m),

(a - 1 a - 1 + ℓ + m) \equiv (a - 1 a - 1 + ℓ) (mod m),

(a - 1 a - 1 + ℓ + m) \equiv (a - 1 a - 1 + ℓ) (mod m),

B_{m}^{(k)} (q) = n = 0 \sum \infty b_{m}^{(k)} (n) q^{n} = j = 0 \prod \infty (1 - q^{m^{j}})^{- k_{j}} .

B_{m}^{(k)} (q) = n = 0 \sum \infty b_{m}^{(k)} (n) q^{n} = j = 0 \prod \infty (1 - q^{m^{j}})^{- k_{j}} .

B_{m}^{(k)} (q) \equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) j = 1 \prod \infty ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m) .

B_{m}^{(k)} (q) \equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) j = 1 \prod \infty ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m) .

P_{i} = j = 0 \prod i (1 - q^{m^{j}})^{- k_{j}}, i \geq 0.

P_{i} = j = 0 \prod i (1 - q^{m^{j}})^{- k_{j}}, i \geq 0.

P_{i} \equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) (1 - q^{m^{i + 1}})^{- 1} j = 1 \prod i ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m),

P_{i} \equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) (1 - q^{m^{i + 1}})^{- 1} j = 1 \prod i ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m),

P_{i + 1}

P_{i + 1}

\equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) (1 - q^{m^{i + 1}})^{- k_{i + 1} - 1} j = 1 \prod i ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m)

\equiv (ℓ_{0} = 0 \sum m - 1 (k _{0} - 1 k _{0} - 1 + ℓ _{0}) q^{ℓ_{0}}) (1 - q^{m^{i + 2}})^{- 1} j = 1 \prod i + 1 ℓ_{j} = 0 \sum m - 1 (k _{j} k _{j} + ℓ _{j}) q^{ℓ_{j} m^{j}} (mod m),

B_{m}^{(k)} (q) = i \to \infty lim P_{i} .

B_{m}^{(k)} (q) = i \to \infty lim P_{i} .

n = d_{0} + d_{1} m + \dots + d_{t} m^{t}, 0 \leq t .

n = d_{0} + d_{1} m + \dots + d_{t} m^{t}, 0 \leq t .

b_{m}^{(k)} (n) \equiv (k _{0} - 1 k _{0} - 1 + d _{0}) j = 1 \prod t (k _{j} k _{j} + d _{j}) (mod m) .

b_{m}^{(k)} (n) \equiv (k _{0} - 1 k _{0} - 1 + d _{0}) j = 1 \prod t (k _{j} k _{j} + d _{j}) (mod m) .

C_{m}^{(k)} (q) = 1 + n = 0 \sum \infty c_{m}^{(k)} (n) q^{n} = 1 + i = 0 \sum \infty j = 0 \prod i ((1 - q^{m^{j}})^{- k_{j}} - 1) .

C_{m}^{(k)} (q) = 1 + n = 0 \sum \infty c_{m}^{(k)} (n) q^{n} = 1 + i = 0 \sum \infty j = 0 \prod i ((1 - q^{m^{j}})^{- k_{j}} - 1) .

C^{({\mathbf{k}})}_{m}(q)\equiv 1+\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\sum_{i=0}^{\infty}\left(1-q^{m^{i+1}}\right)^{-1}\prod_{j=1}^{i}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m).

C^{({\mathbf{k}})}_{m}(q)\equiv 1+\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\sum_{i=0}^{\infty}\left(1-q^{m^{i+1}}\right)^{-1}\prod_{j=1}^{i}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m).

R_{i} = j = 0 \prod i ((1 - q^{m^{j}})^{- k_{j}} - 1), i \geq 0.

R_{i} = j = 0 \prod i ((1 - q^{m^{j}})^{- k_{j}} - 1), i \geq 0.

R_{i}\equiv\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\left(1-q^{m^{i+1}}\right)^{-1}\prod_{j=1}^{i}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m),

R_{i}\equiv\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\left(1-q^{m^{i+1}}\right)^{-1}\prod_{j=1}^{i}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m),

R_{i + 1}

R_{i + 1}

\displaystyle\equiv\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\bigg{\{}\left(1-q^{m^{i+1}}\right)^{-k_{i+1}-1}-\left(1-q^{m^{i+1}}\right)^{-1}\bigg{\}}

\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\;\prod_{j=1}^{i}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m)

\displaystyle\equiv\left(\sum_{\ell_{0}=1}^{m}{k_{0}-1+\ell_{0}\choose k_{0}-1}q^{\ell_{0}}\right)\left(1-q^{m^{i+2}}\right)^{-1}\prod_{j=1}^{i+1}\left(\sum_{\ell_{j}=0}^{m-1}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}q^{\ell_{j}m^{j}}\right)\;({\mathrm{mod}}\;m),

C_{m}^{(k)} (q) = 1 + i = 0 \sum \infty R_{i} .

C_{m}^{(k)} (q) = 1 + i = 0 \sum \infty R_{i} .

n = d_{s} m^{s} + \dots + d_{t} m^{t}, 1 \leq s \leq t,

n = d_{s} m^{s} + \dots + d_{t} m^{t}, 1 \leq s \leq t,

c^{({\mathbf{k}})}_{m}(n-d_{0})\equiv{k_{0}-1-d_{0}\choose k_{0}-1}\left(\varepsilon_{s}+(-1)^{s-1}\bigg{\{}{k_{s}+d_{s}-1\choose k_{s}}-1\bigg{\}}\sum_{i=s}^{t}\prod_{j=s+1}^{i}\bigg{\{}{k_{j}+d_{j}\choose k_{j}}-1\bigg{\}}\right)\;({\mathrm{mod}}\;m),

c^{({\mathbf{k}})}_{m}(n-d_{0})\equiv{k_{0}-1-d_{0}\choose k_{0}-1}\left(\varepsilon_{s}+(-1)^{s-1}\bigg{\{}{k_{s}+d_{s}-1\choose k_{s}}-1\bigg{\}}\sum_{i=s}^{t}\prod_{j=s+1}^{i}\bigg{\{}{k_{j}+d_{j}\choose k_{j}}-1\bigg{\}}\right)\;({\mathrm{mod}}\;m),

n - d_{0} = m - d_{0} + (m - 1) m^{1} + \dots + (m - 1) m^{s - 1} + (d_{s} - 1) m^{s} + d_{s + 1} m^{s + 1} + \dots + d_{t} m^{t} .

n - d_{0} = m - d_{0} + (m - 1) m^{1} + \dots + (m - 1) m^{s - 1} + (d_{s} - 1) m^{s} + d_{s + 1} m^{s + 1} + \dots + d_{t} m^{t} .

(k _{j} - 1 k _{j} - 1 + ℓ _{j}) - 1 = (k _{0} - 1 k _{0} - 1 + m - d _{0}) = (k _{0} - 1 k _{0} - 1 - d _{0}), from (\ref bin coe f f e q u i v),

(k _{j} - 1 k _{j} - 1 + ℓ _{j}) - 1 = (k _{0} - 1 k _{0} - 1 + m - d _{0}) = (k _{0} - 1 k _{0} - 1 - d _{0}), from (\ref bin coe f f e q u i v),

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{j} k _{j} - 1) - 1 = 0 - 1 = - 1,

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{j} k _{j} - 1) - 1 = 0 - 1 = - 1,

\sum_{i=0}^{s-1}\prod_{j=1}^{i}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}=\sum_{i=0}^{s-1}(-1)^{i}=\varepsilon_{s},

\sum_{i=0}^{s-1}\prod_{j=1}^{i}\bigg{\{}{k_{j}+\ell_{j}\choose k_{j}}-1\bigg{\}}=\sum_{i=0}^{s-1}(-1)^{i}=\varepsilon_{s},

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{s} k _{s} + d _{s} - 1) - 1,

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{s} k _{s} + d _{s} - 1) - 1,

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{j} k _{j} + d _{j}) - 1,

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{j} k _{j} + d _{j}) - 1,

(k _{j} k _{j} + ℓ _{j}) - 1 = (k _{j} k _{j}) - 1 = 1 - 1 = 0.

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TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Analytic Number Theory Research

Full text

Characterizing the number of coloured $m-$ ary partitions modulo $m$ , with and without gaps

I. P. Goulden

Dept. of Combinatorics and Optimization, University of Waterloo, Canada

[email protected], [email protected]

and

Pavel Shuldiner

Abstract.

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of $m$ -ary partitions modulo $m$ , with and without gaps. In this paper we extend these results to the case of coloured $m$ -ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo $m$ .

Key words and phrases:

partition, congruence, generating function

2010 Mathematics Subject Classification:

Primary 05A17, 11P83; Secondary 05A15

The work of IPG was supported by an NSERC Discovery Grant.

1. Introduction

An $m$ -ary partition is an integer partition in which each part is a nonnegative integer power of a fixed integer $m\geq 2$ . An $m$ -ary partition without gaps is an $m$ -ary partition in which $m^{j}$ must occur as a part whenever $m^{j+1}$ occurs as a part, for every nonnegative integer $j$ .

Recently, Andrews, Fraenkl and Sellers [AFS15] found an explicit expression that characterizes the number of $m$ -ary partitions of a nonnegative integer $n$ modulo $m$ ; remarkably, this expression depended only on the coefficients in the base $m$ representation of $n$ . Subsequently Andrews, Fraenkel and Sellers [AFS16] followed this up with a similar result for the number of $m$ -ary partitions without gaps, of a nonnegative integer $n$ modulo $m$ ; again, they were able to obtain a (more complicated) explicit expression, and again this expression depended only on the coefficients in the base $m$ representation of $n$ . See also Edgar [E16] and Ekhad and Zeilberger [EZ15] for more on these results.

The study of congruences for integer partition numbers has a long history, starting with the work of Ramanujan (see, e.g., [R19]). For the special case of $m$ -ary partitions, a number of authors have studied congruence properties, including Churchhouse [C69] for $m=2$ , Rødseth [R70] for $m$ a prime, and Andrews [A71] for arbitrary positive integers $m\geq 2$ . The numbers of $m$ -ary partitions without gaps had been previously considered by Bessenrodt, Olsson and Sellers [BOS13] for $m=2$ .

In this note, we consider $m$ -ary partitions, with and without gaps, in which the parts are coloured. To specify the number of colours for parts of each size, we let ${\mathbf{k}}=(k_{0},k_{1},\ldots)$ for positive integers $k_{0},k_{1},\ldots$ , and say that an $m$ -ary partition is ${\mathbf{k}}$ -coloured when there are $k_{j}$ colours for the part $m^{j}$ , for $j\geq 0$ . This means that there are $k_{j}$ different kinds of parts of the same size $m^{j}$ . Let $b^{({\mathbf{k}})}_{m}(n)$ denote the number of ${\mathbf{k}}$ -coloured $m$ -ary partitions of $n$ , and let $c^{({\mathbf{k}})}_{m}(n)$ denote the number of ${\mathbf{k}}$ -coloured $m$ -ary partitions of $n$ without gaps. For the latter, some part $m^{j}$ of any colour must occur as a part whenever some part $m^{j+1}$ of any colour (not necessarily the same colour) occurs as a part, for every nonnegative integer $j$ .

We extend the results of Andrews, Fraenkel and Sellers in [AFS15] and [AFS16] to the case of ${\mathbf{k}}$ -coloured $m$ -ary partitions, where $m$ is relatively prime to $(k_{0}-1)!$ and to $k_{j}!$ for $j\geq 1$ . Our method of proof is different, giving explicit expansions for the generating functions modulo $m$ . These expansions depend on the following simple result.

Proposition 1.1.

For positive integers $m,a$ with $m$ relatively prime to $(a-1)!$ , we have

[TABLE]

Proof.

From the binomial theorem we have

[TABLE]

Now using the falling factorial notation $(a-1+\ell)_{a-1}=(a-1+\ell)(a-2+\ell)\cdots(1+\ell)$ we have

[TABLE]

But

[TABLE]

for any integer $\ell$ , and $\left((a-1)!\right)^{-1}$ exists in $\mathbb{Z}_{m}$ since $m$ is relatively prime to $(a-1)!$ , which gives

[TABLE]

and the result follows. ∎

2. Coloured $m$ -ary partitions

In this section we consider the following generating function for the numbers $b^{({\mathbf{k}})}_{m}(n)$ of ${\mathbf{k}}$ -coloured $m$ -ary partitions:

[TABLE]

The following result gives an explicit expansion for $B^{({\mathbf{k}})}_{m}(q)$ modulo $m$ .

Theorem 2.1.

If $m$ is relatively prime to $(k_{0}-1)!$ and to $k_{j}!$ for $j\geq 1$ , then we have

[TABLE]

Proof.

Consider the finite product

[TABLE]

We prove that

[TABLE]

by induction on $i$ . As a base case, the result for $i=0$ follows immediately from Proposition 1.1 with $a=k_{0}$ . Now assume that (2) holds for some choice of $i\geq 0$ , and we obtain

[TABLE]

where the second last equivalence follows from the induction hypothesis, and the last equivalence follows from Proposition 1.1 with $a=k_{i+1}+1$ , $q=q^{m^{i+1}}$ .

This completes the proof of (2) by induction on $i$ , and the result follows immediately since

[TABLE]

∎

Now we give the explicit expression for the coefficients modulo $m$ that follows from the above expansion of the generating function $B^{({\mathbf{k}})}_{m}(q)$ .

Corollary 2.2.

For $n\geq 0$ , suppose that the base $m$ representation of $n$ is given by

[TABLE]

If $m$ is relatively prime to $(k_{0}-1)!$ and to $k_{j}!$ for $j\geq 1$ , then we have

[TABLE]

Proof.

In the expansion of the series $B^{({\mathbf{k}})}_{m}(q)$ given in Theorem 2.1, the monomial $q^{n}$ arises uniquely with the specializations $\ell_{j}=d_{j}$ , $j=0,\ldots,t$ and $\ell_{j}=0$ , $j\geq t$ . But with these specializations, we have ${k_{j}+\ell_{j}\choose k_{j}}={k_{j}\choose k_{j}}=1$ , and the result follows immediately. ∎

Specializing the expression given in Corollary 2.2 to the case $k_{j}=1$ for $j\geq 0$ provides an alternative proof to Andrews, Fraenkel and Sellers’ characterization of $m-$ ary partitions modulo $m$ , which was given as Theorem 1 of [AFS15].

3. Coloured $m$ -ary partitions without gaps

In this section we consider the following generating function for the numbers $c^{({\mathbf{k}})}_{m}(n)$ of ${\mathbf{k}}$ -coloured $m$ -ary partitions without gaps:

[TABLE]

The following result gives an explicit expansion for $C^{({\mathbf{k}})}_{m}(q)$ modulo $m$ .

Theorem 3.1.

If $m$ is relatively prime to $(k_{0}-1)!$ and to $k_{j}!$ for $j\geq 1$ , then we have

[TABLE]

Proof.

Consider the finite product

[TABLE]

We prove that

[TABLE]

by induction on $i$ . As a base case, the result for $i=0$ follows immediately from Proposition 1.1 with $a=k_{0}$ . Now assume that (3) holds for some choice of $i\geq 0$ , and we obtain

[TABLE]

where the second last equivalence follows from the induction hypothesis, and the last equivalence follows from Proposition 1.1 with $a=k_{i+1}+1$ , $q=q^{m^{i+1}}$ and $a=1$ , $q=q^{m^{i+1}}$ .

This completes the proof of (3) by induction on $i$ , and the result follows immediately since

[TABLE]

∎

Corollary 3.2.

For $n\geq 1$ , suppose that $n$ is divisible by $m$ , with base $m$ representation given by

[TABLE]

where $1\leq d_{s}\leq m-1$ , and $0\leq d_{s+1},\ldots,d_{t}\leq m-1$ . If $m$ is relatively prime to $(k_{0}-1)!$ and to $k_{j}!$ for $j\geq 1$ , then for $0\leq d_{0}\leq m-1$ we have

[TABLE]

where $\varepsilon_{s}=0$ if $s$ is even, and $\varepsilon_{s}=1$ if $s$ is odd.

Proof.

First note that we have

[TABLE]

Now consider the following specializations: $\ell_{0}=m-d_{0}$ , $\ell_{j}=m-1$ , $j=1,\ldots,s-1$ , $\ell_{s}=d_{s}-1$ , $\ell_{j}=d_{j}$ , $j=s+1,\ldots,t$ , and $\ell_{j}=0$ , $j>t$ . Then, in the expansion of the series $C^{({\mathbf{k}})}_{m}(q)$ given in Theorem 3.1, the monomial $q^{n}$ arises once for each $i\geq 0$ , in particular with the above specializations truncated to $\ell_{0},\ldots,\ell_{i}$ . But with these specializations we have

•

for $j=0$ :

[TABLE]

•

for $j=1,\ldots,s-1$ :

[TABLE]

and

[TABLE]

•

for $j=s$ :

[TABLE]

•

for $j=s+1,\ldots,t$ :

[TABLE]

•

for $j>t$ :

[TABLE]

The result follows straightforwardly from Theorem 3.1. ∎

Specializing the expression given in Corollary 3.2 to the case $k_{j}=1$ for $j\geq 0$ provides an alternative proof to Andrews, Fraenkel and Sellers’ characterization of $m-$ ary partitions modulo $m$ without gaps, which was given as Theorem 2.1 of [AFS16].

Bibliography9

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[A 71] G. E. Andrews, Congruence properties of the m 𝑚 m -ary partition function , J. Number Theory 3 (1971), 104–110.
2[AFS 15] George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, Characterizing the Number of m-ary Partitions Modulo m , American Mathematical Monthly 122 (2015), 880–885.
3[AFS 16] George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, m-ary partitions with no gaps: A characterization modulo m , Discrete Mathematics 339 (2016), 283–287.
4[BOS 13] C. Bessenrodt, J. B. Olsson, J. A. Sellers, Unique path partitions: characterization and congruences , Annals Comb. 17 (2013), 591–602.
5[C 69] R. F. Churchhouse, Congruence properties of the binary partition function , Proc. Cambridge Philos. Soc. 66 (1969), 371–376.
6[E 16] Tom Edgar, The distribution of the number of parts of m 𝑚 m -ary partitions modulo m 𝑚 m , Rocky Mountain J. Math. (to appear), ar Xiv 1603.00085 math.CO
7[EZ 15] Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m 𝑚 m -ary partitions mod m 𝑚 m (and doing MUCH more!) , ar Xiv 1511.06791 math.CO
8[R 19] S. Ramanujan, Some properties of p ( n ) 𝑝 𝑛 p(n) , the number of partitions of n 𝑛 n , Proc. Cambridge Philos. Soc. 19 (1919), 207–210.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Taxonomy

Characterizing the number of coloured m−m-m−ary partitions modulo mmm, with and without gaps

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Proposition 1.1**.**

Proof.

2. Coloured mmm-ary partitions

Theorem 2.1**.**

Proof.

Corollary 2.2**.**

Proof.

3. Coloured mmm-ary partitions without gaps

Theorem 3.1**.**

Proof.

Corollary 3.2**.**

Proof.

Characterizing the number of coloured $m-$ ary partitions modulo $m$ , with and without gaps

Proposition 1.1.

2. Coloured $m$ -ary partitions

Theorem 2.1.

Corollary 2.2.

3. Coloured $m$ -ary partitions without gaps

Theorem 3.1.

Corollary 3.2.