Combinatorial Proof of the Minimal Excludant Theorem

TL;DR

This paper provides a combinatorial proof for a theorem relating the sum of minimal excludants of partitions to partitions into distinct parts with two colors, and extends the interpretation to a generalized concept involving least r-gaps.

Contribution

It offers a purely combinatorial proof of the minimal excludant theorem and introduces a generalization involving least r-gaps in partitions.

Findings

Established a combinatorial proof for the minimal excludant sum theorem.

Extended the interpretation to least r-gaps in partitions.

Derived new properties of the function σ_mex(n).

Abstract

The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the smallest positive integer that is not a part of $\lambda$. For a positive integer $n$, $ \sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $\sigma\, \rm{mex}(n)$. They showed, using generating functions, that $\sigma\, \rm{mex}(n)$ equals the number of partitions of $n$ into distinct parts using two colors. In this paper, we provide a purely combinatorial proof of this result and new properties of the function $\sigma\, \rm{mex}(n)$. We generalize this combinatorial interpretation to $\sigma_r\, \rm{mex}(n)$, the sum of least $r$-gaps in all partitions of $n$. The least $r$-gap of a partition $\lambda$ is the smallest positive integer that does not appear at least $r$ times as a part of $\lambda$.

Equations

$$\sigma \text{mex}(n):=\sum _{\lambda \in \mathcal{P}(n)}\text{mex}(\lambda ),$$

$$\sigma \mathrm{mex}(n)=D_2(n).$$

$$\sigma \text{mex}(n)=\sum _{k⩾0}p(n-k(k+1)/2),$$

$$\sigma\,\mathrm{mex}(n)-\sigma\,\mathrm{mex}(n-1)-\delta(n)=\sum_{j=0}^{\infty}M_{1}\big{(}n-j(j+1)/2\big{)},$$

$$\delta (n)=\{\begin{array}{ll}1,& \text{if n=m(m+1)/2, m∈Z,}\\ 0,& \text{otherwise.}\end{array}$$

$$\displaystyle(-1)^{k-1}\left(\sum_{j=-(k-1)}^{k}(-1)^{j}\sigma\,\mathrm{mex}\big{(}n-j(3j-1)/2\big{)}-\delta(n)\right)$$

$$\displaystyle=\sum_{j=0}^{\infty}M_{k}\big{(}n-j(j+1)/2\big{)}.$$

$$\displaystyle(-1)^{k-1}\left(\sum_{j=-(k-1)}^{k}(-1)^{j}\sigma\,\mathrm{mex}\big{(}n-j(3j-1)/2\big{)}-\delta(n)\right)\geqslant 0,$$

$$\displaystyle\sum_{j=0}^{\infty}M_{k}\big{(}n-j(j+1)/2\big{)}$$

$$\text{yng}(3,5,8,4,4,2,2,1)$$

$$\phi :\bigcup _{k⩾0}\mathcal{P}(n-k(k+1)/2)\to {\mathcal{D}}_2(n)$$

$$\frac{(-1{)}^{k-1}}{(q;q{)}_{\infty }}\sum _{n=-(k-1)}^k(-1{)}^j{q}^{n(3n-1)/2}=(-1{)}^{k-1}+\sum _{n=k}^{\infty }\frac{q^{\left(\genfrac{}{}{0ex}{}{k}{2}\right)+(k+1)n}}{(q;q{)}_n}\left[\begin{array}{c}n-1\\ k-1\end{array}\right],$$

$$(a;q{)}_n=\{\begin{array}{ll}1,& \text{if n=0,}\\ \prod _{k=0}^{n-1}(1-a{q}^k),& \text{otherwise,}\end{array}$$

$$\left[\begin{array}{c}n\\ k\end{array}\right]=\{\begin{array}{ll}\frac{(q;q{)}_n}{(q;q{)}_k(q;q{)}_{n-k}},& \text{if 0⩽k⩽n},\\ 0,& \text{otherwise.}\end{array}$$

$$\frac{(q^2;q^2{)}_{\infty }}{(q,q^2{)}_{\infty }}=\sum _{n=0}^{\infty }{q}^{n(n+1)/2},$$

$$\displaystyle(-1)^{k-1}\left(\bigg{(}\sum_{n=0}^{\infty}\sigma\,\mathrm{mex}(n)q^{n}\bigg{)}\bigg{(}\sum_{n=-(k-1)}^{k}(-1)^{j}q^{n(3n-1)/2}\bigg{)}-\sum_{n=0}^{\infty}q^{n(n+1)/2}\right)$$

$$=\left(\sum _{n=0}^{\infty }{q}^{n(n+1)/2}\right)\left(\sum _{n=0}^{\infty }{M}_k(n)q^n\right),$$

$$\sum _{n=0}^{\infty }\sigma \mathrm{mex}(n)q^n=\frac{(q^2;q^2{)}_{\infty }}{(q;q{)}_{\infty }(q;q^2{)}_{\infty }}$$

$$\sum _{n=0}^{\infty }{M}_k(n)q^n=\sum _{n=k}^{\infty }\frac{q^{\left(\genfrac{}{}{0ex}{}{k}{2}\right)+(k+1)n}}{(q;q{)}_n}\left[\begin{array}{c}n-1\\ k-1\end{array}\right].$$

$$\sigma \text{mex}\left(n-\frac{k(3k+1)}{2}\right)-\sigma \text{mex}\left(n-\frac{k(3k+5)}{2}-1\right)$$

$$\displaystyle\qquad\qquad=\sum_{j=0}^{\infty}\Big{(}M_{k}\big{(}n-j(j+1)/2\big{)}+M_{k+1}\big{(}n-j(j+1)/2\big{)}\Big{)}.$$

$$\displaystyle\sum_{j=0}^{\infty}\Bigg{(}p\bigg{(}n-\frac{j(j+1)}{2}-\frac{k(3k+1)}{2}\bigg{)}-p\bigg{(}n-\frac{j(j+1)}{2}-\frac{k(3k+5)}{2}-1\bigg{)}\Bigg{)}$$

$$\displaystyle\qquad\qquad=\sum_{j=0}^{\infty}\Big{(}M_{k}\big{(}n-j(j+1)/2\big{)}+M_{k+1}\big{(}n-j(j+1)/2\big{)}\Big{)}.$$

$$\mathrm{sign}(\mathrm{r})=\{\begin{array}{ll}1& \text{mbox}ifr⩾0\\ -1& \text{mbox}ifr<0.\end{array}$$

$$\sum_{j=0}^{\infty}M_{k}\big{(}n-j(j+1)/2\big{)}=D^{(k)}_{3}(n).$$

$$(-1)^{\max(0,k-1)}\left(\sum_{j=-\max(0,k-1)}^{k}(-1)^{j}\sigma\,\mathrm{mex}\big{(}n-j(3j-1)/2\big{)}-\delta(n)\right)\\ =D^{(k)}_{3}(n).$$

$$3,\overline{3},2+1,\overline{2}+1,2+\overline{1},\overline{2}+\overline{1},1+1+1,\overline{1}+1+1.$$

$$(-1{)}^k\left(\sigma \mathrm{mex}(n)+2\sum _{j=1}^k(-1{)}^j\sigma \mathrm{mex}(n-j^2)-{\delta }^{\prime }(n)\right)$$

$$\displaystyle=\sum_{j=-\infty}^{\infty}(-1)^{j}\overline{M}_{k}\big{(}n-j(3j-1)\big{)},$$

$${\delta }^{\prime }(n)=\{\begin{array}{ll}(-1{)}^m,& \text{if n=m(3m−1), m∈Z,}\\ 0,& \text{otherwise.}\end{array}$$

$$\frac{(-q;q{)}_{\infty }}{(q;q{)}_{\infty }}\left(1+2\sum _{j=1}^k(-1{)}^j{q}^{j^2}\right)$$

$$=1+2(-1{)}^k\frac{(-q;q{)}_k}{(q;q{)}_k}\sum _{j=0}^{\infty }\frac{q^{(k+1)(k+j+1)}(-q^{k+j+2};q{)}_{\infty }}{(1-q^{k+j+1})(q^{k+j+2};q{)}_{\infty }},$$

$$\sum _{n=0}^{\infty }{\overline{M}}_k(n)q^n=2\frac{(-q;q{)}_k}{(q;q{)}_k}\sum _{j=0}^{\infty }\frac{q^{(k+1)(k+j+1)}(-q^{k+j+2};q{)}_{\infty }}{(1-q^{k+j+1})(q^{k+j+2};q{)}_{\infty }}.$$

Full text

Keywords: Minimal excludant, MEX, least gap in partition, partitions, $2$-color partitions

MSC 2010: 11A63, 11P81, 05A19

1 Introduction

The minimal excludant or mex-function on a set $S$ of positive integers is the least positive integer not in $S$. The history of this notion goes back to at least the 1930s when it was applied to combinatorial game theory [11, 9].

Recently, Andrews and Newman [4] considered the mex-function applied to integer partitions. They defined the minimal excludant of a partition $\lambda$, $\text{mex}(\lambda)$, to be the smallest positive integer that is not a part of $\lambda$. In addition, for each positive integer $n$, they defined

[TABLE]

where $\mathcal{P}(n)$ is the set of all partitions of $n$. Elsewhere in the literature, the minimal excludant of a partition $\lambda$ is referred to as the least gap or smallest gap of $\lambda$. An exact and asymptotic formula for $\sigma\,\text{mex}(n)$ is given in [8]. In [5], where $\text{mex}(\lambda)$ is denoted by $g_{1}(\lambda)$ and $\sigma\,\text{mex}(n)$ is denoted by $S_{1}(n)$, the authors study a generalization of $\sigma\,\text{mex}(n)$ and its connection to polygonal numbers.

Let $\mathcal{D}_{2}(n)$ be the set of partitions of $n$ into distinct parts using two colors and let $D_{2}(n)=|\mathcal{D}_{2}(n)|$. For ease of notation, we denote the colors of the parts of partitions in $\mathcal{D}_{2}(n)$ by [math] and $1$. In [4], the authors give two proofs of the following theorem.

Theorem 1.1**.**

Given an integer $n\geqslant 0$, we have

[TABLE]

They also determine the parity of the $\sigma\,\text{mex}$ function.

Lemma 1.2**.**

$\sigma\,\text{mex}(n)$* is odd if and only if $n=j(3j\pm 1)$ for some non-negative integer $j$.*

We note that this parity result was also established in [5, Corollary 1.6]. Andrews and Newman write that “it would be of great interest to have a bijective proof of Theorem 1.1.” They also ask for a combinatorial proof of Lemma 1.2. In section 2 we provide these desired proofs. In the proof of Theorem 1.1, we make use of the fact that

[TABLE]

where, as usual, $p(n)$ denotes the number of partitions of $n$. A combinatorial proof of (1) is given in [5, Theorem 1.1] . The same argument is also described in the second proof of [4, Theorem 1.1]. We note the result proven in [5] is a generalization of (1) to the sum of $r$-gaps in all partitions of $n$. The $r$-gap of a partition $\lambda$ is the least positive integer that does not appear $r$ times as a part of $\lambda$.

In [2], Andrews and Merca considered a restricted mex-function and defined $M_{k}(n)$ to be the number of partitions of $n$ in which $k$ is the least positive integer that is not a part and there are more parts $>k$ than there are parts $<k$.

When $k=1$, $M_{1}(0)=0$ and, if $n>0$, $M_{1}(n)$ is the number of partitions of $n$ that do not contain $1$ as a part. Thus, if $n>0$, we have $M_{1}(n)=p(n)-p(n-1)$. Then, from $\eqref{smex as sum}$, we obtain

[TABLE]

where $\delta$ is the characteristic function of the set of triangular numbers, i.e.,

[TABLE]

In section 3, we prove the following generalization of (2).

Theorem 1.3**.**

Let $k$ be a positive integer. Given an integer $n\geqslant 0$, we have

[TABLE]

As a corollary of this theorem we obtain the following infinite family of linear inequalities involving $\sigma\,\mathrm{mex}$.

Corollary 1.4**.**

Let $k$ be a positive integer. Given an integer $n\geqslant 0$, we have

[TABLE]

with strict inequality if $n\geqslant k(3k+1)/2$.

In section 3 we also give a combinatorial interpretation for

[TABLE]

in terms of the number of partitions into distinct parts using three colors and satisfying certain conditions.

In sections 4 and 5, we introduce connections to certain subsets of overpartitions and partitions with distinct parts, respectively.

2 Combinatorial Proofs of Theorem 1.1 and Lemma 1.2

2.1 Bijective Proof of Theorem 1.1

To prove the theorem, we adapt Sylvester’s bijective proof of Jacobi’s triple product identity [12] which was later rediscovered by Wright [13]. For the interested reader, it is probably easier to follow Wright’s short article for the description of the bijection.

Given a partition $\lambda$ in $\mathcal{D}_{2}(n)$, let $\lambda^{(j)}$, $j=0,1$, be the (uncolored) partition whose parts are the parts of color $j$ in $\lambda$. Then, $\lambda^{(1)}$ and $\lambda^{(2)}$ are partitions into distinct parts.

Example 2.1**.**

If $\lambda=4_{1}+3_{0}+3_{1}+2_{0}+1_{0}\in\mathcal{D}_{2}(13)$, then $\lambda^{(0)}=3+2+1$ and $\lambda^{(1)}=4+3$.

Denote by $\eta(k)$ the staircase partition $\eta(k)=k+(k-1)+(k-2)+\cdots+3+2+1$. If $k=0$ we define $\eta(k)=\emptyset$. For any partition $\lambda$ we denote by $\ell(\lambda)$ the number of parts in $\lambda$.

Definition 1**.**

Given diagram of left justified rows of boxes (not necessarily the Ferrers diagram of a partition), the staircase profile of the diagram is a zig-zag line starting in the upper left corner of the diagram with a right step and continuing in alternating down and right steps until the end of a row of the diagram is reached.

Example 2.2**.**

The staircase profile of the diagram

[TABLE]

is

\ydiagram

3,5,8,4,4,2,2,1

Given a Ferrers diagram (with boxes of unit length) of a partition $\lambda$ into distinct parts, the shifted Ferrers diagram of $\lambda$ is the diagram in which row $i$ is shifted $i-1$ units to the right.

We create a map

[TABLE]

as follows. Start with $\lambda\in\mathcal{P}(n-k(k+1)/2)$ for some $k\geqslant 0$. Append a diagram with rows of lengths $1,2,\ldots k$ (i.e., the Ferrers diagram of $\eta(k)$ rotated by $180^{\circ}$) the top of Ferrers diagram of $\lambda$. We obtain a diagram with $n$ boxes. Draw the staircase profile of the new diagram. Let $\alpha$ be the partition whose parts are the length of the columns to the left of the staircase profile and let $\beta$ be the partition whose parts are the length of the rows to the right of the staircase profile. Then $\alpha$ and $\beta$ are partitions with distinct parts. Moreover, $k\leqslant\ell(\alpha)-\ell(\beta)\leqslant k+1$. Color the parts of $\alpha$ with color $k\!\!\pmod{2}$ and the parts of $\beta$ with color $k+1\!\!\pmod{2}$. Then $\varphi(\lambda)$ is defined as the 2-color partition of $n$ whose parts are the colored parts of $\alpha$ and $\beta$.

Conversely, start with $\mu\in\mathcal{D}_{2}(n)$. Let $\ell_{j}(\mu)$, $j=0,1$, be the number of parts of color $j$ in $\mu$.

(i) If $\ell_{0}(\mu)\geqslant\ell_{1}(\mu)$, let $r=\ell_{0}(\mu)-\ell_{1}(\mu)$. Let $k=\displaystyle r+\frac{(-1)^{r}-1}{2}$. Remove the top $k$ rows (i.e., the rotated Ferrers diagram of $\eta(k)$) from the conjugate of the shifted diagram of $\mu^{(0)}$ and join the remaining diagram with the shifted digram of $\mu^{(1)}$ so they align at the top. The obtained partition $\varphi^{-1}(\mu)$ belongs to $\mathcal{P}(n-k(k+1)/2)$.

(ii) If $\ell_{1}(\mu)>\ell_{0}(\mu)$, let $r=\ell_{1}(\mu)-\ell_{0}(\mu)$. Let $k=\displaystyle r-\frac{(-1)^{r}+1}{2}$. Remove the top $k$ rows (i.e., the rotated Ferrers diagram of $\eta(k)$) from the conjugate of the shifted diagram of $\mu^{(1)}$ and join the remaining diagram with the shifted digram of $\mu^{(0)}$ so they align at the top. The obtained partition $\varphi^{-1}(\mu)$ belongs to $\mathcal{P}(n-k(k+1)/2)$.

Example 2.3**.**

Let $n=38,k=3$, and let $\lambda=7+7+6+6+4+2$ be a partition of $n-k(k+1)/2=32$. We add the rotated Ferrers diagram of $\eta(3)$ to the top of the Ferrers diagram of $\lambda$ and draw the staircase profile.

\ydiagram

1,2,3,7,7,6,6,4,2

Then $\alpha=9+8+6+5+3+2$ and $\beta=3+2$. Since $k$ is odd, we have $\varphi(\lambda)=9_{1}+8_{1}+6_{1}+5_{1}+3_{1}+3_{0}+2_{1}+2_{0}\in\mathcal{D}_{2}(38)$.

Conversely, suppose $\mu=9_{1}+8_{1}+6_{1}+5_{1}+3_{1}+3_{0}+2_{1}+2_{0}\in\mathcal{D}_{2}(38)$. Then $\ell_{0}(\mu)=2$ and $\ell_{1}(\mu)=6$. We have $r=\ell_{1}(\mu)-\ell_{0}(\mu)=4$ and $k=3$. The diagrams of the conjugate of the shifted diagram of $\mu^{(1)}$ and the shifted diagram of $\mu^{(0)}$ are shown below.

Next, we remove the first $3$ rows from the conjugate of the shifted diagram of $\mu^{(1)}$ and join the remaining diagram and the shifted digram of $\mu^{(0)}$ so they align at the top. We obtain $\varphi^{-1}(\mu)=7+7+6+6+4+2\in\mathcal{P}(32)$.

2.2 Combinatorial Proof of Lemma 1.2

To determine the parity of $\sigma\,\text{mex}(n)$, we pair partitions in $\mathcal{D}_{2}(n)$ as follows. If $\mu\in\mathcal{D}_{2}(n)$, we denote by $\tilde{\mu}$ the partition in $\mathcal{D}_{2}(n)$ obtained by interchanging the colors of the part of $\mu$. Then $\mu\neq\tilde{\mu}$ if and only if $\mu^{(0)}\neq\mu^{(1)}$. If $n$ is odd, $\mu^{(0)}\neq\mu^{(1)}$ for all $\mu\in\mathcal{D}_{2}(n)$ and $\sigma\,\text{mex}$ is even. If $n$ is even, $\sigma\,\text{mex}\equiv q(n/2)\pmod{2}$, where, as usual, $q(m)$ denotes the number of partitions of $m$ with distinct parts. Franklin’s involution used to prove Euler’s Pentagonal Number Theorem provides a pairing of partitions with distinct parts that shows that $q(m)$ is odd if and only if $m$ is a generalized pentagonal number. Thus, $\sigma\,\text{mex}(n)$ is odd if and only if $n$ is twice a generalized pentagonal number.

3 Proofs of Theorem 1.3

Analytic proof of Theorem 1.3.

In [2], the authors considered Euler’s pentagonal number theorem and proved the following truncated form:

[TABLE]

where

[TABLE]

and

[TABLE]

Multiplying both sides of (3) by

[TABLE]

we obtain

[TABLE]

where we have invoked the generating function for $\sigma\,\mathrm{mex}(n)$ [5, 4],

[TABLE]

and the generating function for $M_{k}(n)$ [2],

[TABLE]

The proof follows easily considering Cauchy’s multiplication of two power series.∎

Combinatorial proof of Theorem 1.3.

The statement of Theorem 1.3 is equivalent to identity (2) together with

[TABLE]

Using (1), identity (4) becomes

[TABLE]

Identity (5) was proved combinatorially in [14]. Together with the combinatorial proof of (1), this gives a combinatorial proof of Theorem 1.3. ∎

Next, we give a combinatorial interpretation for \displaystyle\sum_{j=0}^{\infty}M_{k}\big{(}n-j(j+1)/2\big{)}. First, we introduce some notation. Given an integer $r$, let $\rm{sign}(r)$ denote the sign of $r$, i.e.

[TABLE]

For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we denote by $D^{(k)}_{3}(n)$ the number of partitions $\mu$ of $n$ into distinct parts using three colors, $0,1,2$, and satisfying the following conditions:

1. (i)

$\mu$ has exactly $k$ parts of color $2$ and, if $k>1$, twice the smallest part of color $2$ is greater than largest part of color $2$. 2. (ii)

Let $r=\ell_{0}(\mu)-\ell_{1}(\mu)$ be the signed difference in the number of parts colored [math] and the number of parts colored $1$ in $\mu$. Let $j=\displaystyle|r|-\frac{1}{2}+\rm{sign}(r)\frac{(-1)^{r}}{2}$. The largest part of color $j\!\!\pmod{2}$ must equal $j$ more that the smallest part of color $2$.

Then, we have the following proposition.

Proposition 3.1**.**

For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we have

[TABLE]

Proof.

Take a partition counted by M_{k}\big{(}n-j(j+1)/2\big{)} and consider its Ferrers diagram. Remove the first $k$ columns and color the length of each of these columns with color $2$. To the remaining Ferrers diagram, add the rotated Ferrers diagram of a staircase $\eta(j)$ of height $j$ and perform the transformation in the combinatorial proof of Theorem 1.1. It is now straight forward that this transformation is a bijection between the sets of partitions counted by the two sides of (6).∎

Combining Theorems 1.1 and 1.3, and Proposition 3.1 we obtain the following corollary which, by the discussion above, has both analytic and combinatorial proofs.

Corollary 3.2**.**

For integers $k,n$ such that $k\geqslant 1$ and $n\geqslant 0$, we have

[TABLE]

Note that, if $k=0$, the statement of the corollary reduces to Theorem 1.1.

4 Connections with overpartitions

Overpartitions are ordinary partitions with the added condition that the first appearance of any part may be overlined or not. There are eight overpartitions of $3$:

[TABLE]

In [3], the authors denoted by $\overline{M}_{k}(n)$ the number of overpartitions of $n$ in which the first part larger than $k$ appears at least $k+1$ times. For example, $\overline{M}_{2}(12)=16$, and the partitions in question are $4+4+4$, $\overline{4}+4+4$, $3+3+3+3$, $\overline{3}+3+3+3$, $3+3+3+2+1$, $3+3+3+\overline{2}+1$, $3+3+3+2+\overline{1}$, $3+3+3+\overline{2}+\overline{1}$, $\overline{3}+3+3+2+1$, $\overline{3}+3+3+\overline{2}+1$, $\overline{3}+3+3+2+\overline{1}$, $\overline{3}+3+3+\overline{2}+\overline{1}$, $3+3+3+1+1+1$, $3+3+3+\overline{1}+1+1$, $\overline{3}+3+3+1+1+1$, $\overline{3}+3+3+\overline{1}+1+1$.

We have the following identity.

Theorem 4.1**.**

For integers $k,n>0$, we have

[TABLE]

where

[TABLE]

Proof.

According to [3, Theorem 7], we have

[TABLE]

where

[TABLE]

Multiplying both sides of (7) by

[TABLE]

we obtain

[TABLE]

and the proof follows easily. ∎

Related to Theorem 4.1, we remark that there is a substantial amount of numerical evidence to conjecture the following inequality.

Conjecture 1**.**

For $k,n>0$,

[TABLE]

with strict inequality if $n\geqslant(k+1)^{2}$.

It would be very appealing to have a combinatorial interpretation for the sum in this conjecture.

5 Connections with partitions into distinct parts

Following the notation for the number of partitions of $n$ into distinct parts of two colors, we denote by $D_{1}(n)$ the number of partitions of $n$ into distinct parts. We prove the following identity.

Theorem 5.1**.**

For any integer $n\geqslant 0$, we have

[TABLE]

where $D_{1}(x)=0$ if $x$ is not a positive integer.

Proof.

Considering the classical theta identity [1, p. 23, eq. (2.2.13)]

[TABLE]

we can write

[TABLE]

and the proof follows by equating the coefficients of $q^{n}$ in this identity. ∎

To obtain a combinatorial interpretation for the sum on the right hand side of (8), let $D_{2}^{*}(n)$ be the number of partitions of $n$ with distinct parts using two colors such that: (i) parts of color [math] form a gap-free partition (staircase) and (ii) only even parts can have color $1$. Then, we have the following identity of Watson type [6].

Proposition 5.2**.**

For $n\geqslant 0$,

[TABLE]

Proof.

To see this, let $\lambda$ be a partition counted by $\displaystyle D_{1}\left(\frac{n-j(j+1)/2}{2}\right)$. Double the size of each part of $\lambda$ to obtain a partition $\mu$ of $n-j(j+1)/2$ whose parts are even and distinct. Color the parts of $\mu$ with color $1$ and add parts $1,2,\ldots,j$ in color [math] to obtain a partition counted by $D_{2}^{*}(n)$. This transformation is clearly reversible. ∎

In [3], the authors denoted by $MP_{k}(n)$ the number of partitions of $n$ in which the first part larger than $2k-1$ is odd and appears exactly $k$ times. All other odd parts appear at most once. For example, $MP_{2}(19)=10$, and the partitions in question are $9+9+1$, $9+5+5$, $8+5+5+1$, $7+7+3+2$, $7+7+2+2+1$, $7+5+5+2$, $6+5+5+3$, $6+5+5+2+1$, $5+5+3+2+2+2$, $5+5+2+2+2+2+1$.

We remark the following truncated form of Theorem 5.1.

Theorem 5.3**.**

For integers $n,k>0$,

[TABLE]

Proof.

According to [3, Theorem 9], we have

[TABLE]

where

[TABLE]

The proof follows easily by multiplying both sides of (10) by

[TABLE]

∎

A further interesting corollary of Theorem 5.3 relates to $\sigma\,\mathrm{mex}(n)$.

Corollary 5.4**.**

For integers $n,k>0$,

[TABLE]

with strict inequality if $n\geqslant k(2k+1)$.

On the other hand, the reciprocal of the infinite product in (9) is the generating function for $\mathrm{pod}(n)$, the number of partitions of $n$ in which odd parts are not repeated, i.e.,

[TABLE]

The properties of the partition function $\mathrm{pod}(n)$ were studied in [10] by Hirschhorn and Sellers. We easily deduce the following convolution identity.

Corollary 5.5**.**

For $n\geqslant 0$,

[TABLE]

Finally, we remark that finding a combinatorial interpretation for

[TABLE]

would be very desirable.

6 Concluding remarks

The present work began with the search for a combinatorial proof of Theorem 1.1. We were further able to prove several truncated series formulas involving the function $\sigma\,\text{mex}$. In [5], we worked with the generalization of this function: the sum, $S_{r}(n)$, of $r$-gaps in all partitions of $n$. To keep notation uniform, we use $\sigma\,\text{mex}_{r}(n)$ for $S_{r}(n)$. Recall that the $r$-gap of a partition $\lambda$ is the least positive integer that does not appear at least $r$ times as a part of $\lambda$. In [5], we proved combinatorially that

[TABLE]

and we gave the generating function for $\sigma_{r}\,\text{mex}(n)$, namely

[TABLE]

Denote by $\widetilde{D}_{2}^{(r)}(n)$ the number of partitions $\lambda$ of $n$ using two colors, [math] and $1$, such that:

1. (i)

$\lambda^{(0)}$ is a partition into distinct parts divisible by $r$. 2. (ii)

$\lambda^{(1)}$ is a partition with parts repeated at most $2r-1$ times.

The following generalization of Theorem 1.1 is immediate from (13).

Theorem 6.1**.**

Let $n,r$ be integers with $r>0$ and $n\geq 0$. Then $\sigma_{r}\,\text{mex}(n)=\widetilde{D}_{2}^{(r)}(n).$

Combinatorial proof of Theorem 6.1.

Let $\widetilde{\mathcal{D}}_{2}^{(r)}(n)$ be the set of partitions of $n$ counted by $\widetilde{D}_{2}^{(r)}(n)$ described above. Let $\mathcal{P}_{r}(n)$ be the set of partitions of $n$ in which all parts are divisible by $r$. Let $\overline{\mathcal{P}}_{r}(n)$ be the set of partitions of $n$ in which all parts are not divisible by $r$. Finally, let $\mathcal{Q}_{r}(n)$ be the set of partitions of $n$ with parts repeated at most $r-1$ times.

Let $\psi$ denote Glaisher’s bijection from $\overline{\mathcal{P}}_{r}(n)$ to $\mathcal{Q}_{r}(n)$.

We create a bijection

[TABLE]

Start with a partition $\lambda\in\mathcal{P}(n-rj(j+1)/2)$ for some $j\geqslant 0$. Let $\tilde{\lambda}$ be the partition consisting of the parts of $\lambda$ that are divisible by $r$ and $\bar{\lambda}$ be the partition consisting of the remaining parts of $\lambda$. Thus all parts of $\bar{\lambda}$ are not divisible by $r$.

Let $\tilde{\lambda}_{/r}$ be the partition obtained from $\tilde{\lambda}$ by dividing each part by $r$. To $\tilde{\lambda}_{/r}\in\mathcal{P}\left(\frac{n-|\bar{\lambda}|}{r}-\frac{j(j+1)}{2}\right)$ we apply the bijection $\varphi$ from the combinatorial proof of Theorem 1.1 in section 2. (The appended rotated staircase is $\eta(j)$.) Then $\varphi(\tilde{\lambda}_{/r})\in\mathcal{D}_{2}\left(\frac{n-|\bar{\lambda}|}{r}\right)$. In $\varphi(\tilde{\lambda}_{/r})$, multiply each part of color [math] by $r$ and repeat each part of color $1$ exactly $r$ times. These parts, together with the parts of $\psi(\bar{\lambda})$ colored $1$, form the partition $\xi(\lambda)\in\widetilde{\mathcal{D}}_{2}^{(r)}(n)$.

Conversely, let $\mu\in\widetilde{\mathcal{D}}_{2}^{(r)}(n)$. Then $\mu^{(0)}$ is a partition with distinct parts all of which are multiples of $r$ and $\mu^{(1)}$ is a partition with parts repeated at most $2r-1$ times. We write $\mu^{(1)}$ as $\mu^{(1)}=\alpha^{(1)}\cup\beta^{(1)}$, where all parts of $\alpha^{(1)}$ have multiplicity exactly $r$ and all parts of $\beta^{(1)}$ have multiplicity at most $r-1$. Then $\psi^{-1}(\beta^{(1)})$ has no part divisible by $r$. We have $|\mu^{(0)}|=rt_{1}$, $|\alpha^{(1)}|=rt_{2}$ and $|\beta^{(1)}|=n-rt_{1}-rt_{2}$ for some non-negative integers $t_{1}$ and $t_{2}$.

Let $\mu^{(0)}_{/r}$ be the partition with parts colored [math] obtained from $\mu^{(0)}$ by dividing each part by $r$. Then, $\mu^{(0)}_{/r}$ is a partition with distinct parts colored [math]. Let $\alpha^{(1)}_{\backslash r}$ be the partition with distinct parts colored $1$ with exactly the same set of parts as $\alpha^{(1)}$. We then apply $\varphi^{-1}$ to $\mu^{(0)}_{/r}\cup\alpha^{(1)}_{\backslash r}\in\mathcal{D}_{2}(t_{1}+t_{2})$ to obtain $\varphi^{-1}(\mu^{(0)}_{/r}\cup\alpha^{(1)}_{\backslash r})\in\mathcal{P}\left(t_{1}+t_{2}-\frac{j(j+1)}{2}\right)$ for some non-negative integer $j$. We multiply each part of $\varphi^{-1}(\mu^{(0)}_{/r}\cup\alpha^{(1)}_{\backslash r})$ by $r$. These parts, together with the parts of $\psi^{-1}(\beta^{(1)})$, form the partition $\xi^{-1}(\mu)\in\mathcal{P}(n-rj(j+1)/2)$.

∎

Example 6.1**.**

Let $n=167,r=3,j=3$ and consider

[TABLE]

Then,

[TABLE]

[TABLE]

and

[TABLE]

Applying Glaisher’s bijection, we have $\psi(\bar{\lambda})=24+19+8+1+1\in\mathcal{Q}_{3}(53)$. From Example 2.3, we have $\varphi(\tilde{\lambda}_{/3})=9_{1}+8_{1}+6_{1}+5_{1}+3_{1}+3_{0}+2_{1}+2_{0}$. Now, we multiply parts of color [math] by $3$, repeat each part of color $1$ three times, and include the parts of $\psi(\bar{\lambda})$ with color $1$ to obtain

[TABLE]

Conversely, let

[TABLE]

Then, we have the following relevant partitions.

[TABLE]

Then $\mu^{(0)}_{/3}\cup\alpha^{(1)}_{\backslash 3}\in\mathcal{D}_{2}(38)$ and from Example 2.3, we have $j=3$ and

[TABLE]

Now we multiply all parts of $\varphi^{-1}(\mu^{(0)}_{/3}\cup\alpha^{(1)}_{\backslash 3})$ by $3$ and include the parts of $\psi^{-1}(\beta^{(1)})$ with the color removed to obtain

[TABLE]