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4 votes
2 answers
343 views

Number of partitions of $n$ and number of different integers in 1-avoiding partitions

Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by $$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$ I have encountered an interesting enumeration. Take ...
T. Amdeberhan's user avatar
2 votes
0 answers
79 views

Skewed plane partition with only row fillings reversed

The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
Zhi Wang's user avatar
3 votes
1 answer
215 views

Seeking for a combinatorial argument for partition identities

Given an integer partition $\lambda$, introduce the following quantities: \begin{align*} c(\lambda)&=\sum_{i\geq1}\left\lceil\frac{\lambda_i}2\right\rceil, \qquad c_o(\lambda)=\sum_{i\geq1}\left\...
T. Amdeberhan's user avatar
6 votes
2 answers
432 views

Plane partitions as sums of determinants

Consider the Vandermonde's determinant computed by $$V(x_1,\dots,x_m):=\det(x_j^{i-1})_{i,j=1}^m=\prod_{1\leq i<j\leq m}(x_i-x_j).$$ The number of plane partitions in an $n\times m\times m$ box (...
T. Amdeberhan's user avatar
0 votes
1 answer
132 views

Seeking a bijective proof enumerating two partition sets: Part II

An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a ...
T. Amdeberhan's user avatar
2 votes
1 answer
181 views

Seeking a bijective proof enumerating two partition sets: Part I

An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a ...
T. Amdeberhan's user avatar
4 votes
1 answer
349 views

The fraction $\frac{g_{\mu}}{f_{\lambda}}$ is an integer

Let $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{\ell(\lambda)}>0)$ be an integer partition of $n\in\mathbb{N}$; i.e., $\lambda_1+\cdots+\lambda_{\ell(\lambda)}=n$. One may now associate $...
T. Amdeberhan's user avatar
12 votes
1 answer
594 views

Equality of two $q$-series. Proof?

Recall the notation $(z;q)_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$. My earlier MO question did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. ...
T. Amdeberhan's user avatar
3 votes
1 answer
220 views

Agreement between two sets of data on partitions

Let $\lambda=(\lambda_1,\lambda_2,\dotsc,\lambda_{\ell(\lambda)})$ be an integer partition of positive numbers where $\ell(\lambda)$ is the length of the partition. One may associate a Ferrer diagram ...
T. Amdeberhan's user avatar
4 votes
1 answer
263 views

A refinment of Beck's conjecture

Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. ...
T. Amdeberhan's user avatar
1 vote
0 answers
203 views

Generalizing "partition into odd parts=partition into distinct parts"?

The number of partitions into distinct parts is known to agree with the number of partitions with odd parts. For instance, this follows from $$\prod_{k=1}^{\infty}(1+q^k)=\prod_{n=1}^{\infty}\frac1{1-...
T. Amdeberhan's user avatar
0 votes
0 answers
75 views

Objects equinumerous with $3$-ary partitions?

There is a concept of the so-called RP-compositions of an integer discussed by K. Q. Ji and H. S. Wilf in Extreme palindromes. They proved the following result too: Theorem. The number of RP-...
T. Amdeberhan's user avatar
13 votes
2 answers
803 views

Two interpretations of a sequence: an opportunity for combinatorics

The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function $$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$ In particular, look ...
T. Amdeberhan's user avatar
1 vote
0 answers
90 views

Dimension of a certain space of symmetric functions: Part II

This is the second installment of my earlier MO question. Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
T. Amdeberhan's user avatar
4 votes
0 answers
205 views

Dimension of a certain space of symmetric functions: Part I

Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$. QUESTION. Consider the ...
T. Amdeberhan's user avatar