All Questions
Tagged with co.combinatorics partitions
300 questions
3
votes
0
answers
73
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
1
vote
1
answer
134
views
A distributive identity for products of partition functions
An $r$-composition of a non-negative integer $s$ is an expression $s=s_1+s_2+\cdots+s_r$ where the $s_i$ are also non-negative integers. Define $k(r,s):=\sum \pi(s_1)\pi(s_2) \cdots \pi(s_r)$ where ...
- 161
2
votes
1
answer
431
views
Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 43.2k
2
votes
1
answer
100
views
Clique number and a special partition
Let $G=(V,E)$ be a finite, simple, undirected, connected graph, and let $\omega(G)$ denote its clique number. Assume that $G$ has a partition into $m$ independent subsets $U_1,\dots, U_m$ such that ...
- 21
4
votes
1
answer
195
views
Optimal partition of $n$ points
Given an integer $n$, and 3 real sequences $\{x_1, \dots, x_n\}, \{y_1, \dots, y_n\}$ and $\{w_1, \dots, w_n\}$ with $x_k, y_k, w_k > 0$, for all $k \in \{1, \dots, n\}$. For a fixed $p < n$ ...
- 43
1
vote
0
answers
118
views
Can we construct the circular permutation from partial partition info?
Imagine a circular permutation of n points on a circle, if we draw a line connecting any pair of points, the rest of the points are divided into two sets that are on the same side. We can partition a ...
- 11
6
votes
0
answers
171
views
An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
3
votes
0
answers
213
views
A family of polynomials related to integer partitions
For a positive integer $n$, let $p(n)$ be the number of partitions of $n$.
For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
- 15.6k
3
votes
1
answer
159
views
Proving that two sequences of polynomials defined over partitions are inverse to each other
For any fixed $c>0$ consider the polynomials
\begin{align*}
& p_n(X_1,X_2,\ldots) := \frac{n!}{c} \sum\limits_{b=1}^n \frac{c^b}{b!(n+1-b)!} \sum\limits_{\substack{l_1,\ldots,l_b \geq 1 \\ ...
- 1,295
10
votes
1
answer
625
views
Generating function for A261041
Let $a(n)$ be A261041 (i.e., number of partitions of subsets of $\{1,2,\dotsc,n\}$, where consecutive integers are required to be in different parts).
Let $b(n)$ be an integer sequence with generating ...
- 4,938
10
votes
1
answer
207
views
Generating function for A225114
Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns).
Let $b(n)$ be an integer sequence with generating function $B(x)$ such that
$$
B(x) = \...
- 4,938
8
votes
0
answers
260
views
Efficient listing of ASMs
Famously, the alternating sign matrix theorem gives a product formula for the number $a(n)$ of ASMs of size $n$. There are multiple proofs of this formula, all somewhat involved. My question is ...
- 17k
4
votes
1
answer
173
views
Approximating a partition
Let $X = \{1,\ldots,2 \cdot n\}$ be a set of numbers. For this question, an equal partition $Y_1,Y_2$ of $X$ is a partition of $X$ to two equal sizes. Let $\varepsilon \in (0,1)$ be an error parameter....
- 173
2
votes
1
answer
217
views
Number of distinct higher dimensional integer partitions
By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
- 281
6
votes
1
answer
337
views
Number of ways a positive integer n can be expressed as a sum of k natural numbers under a certain ordering condition
I have a question that I can't solve for the moment. Suppose we have a fixed positive integer $k$, now consider $k$
natural numbers $x_1,x_2,\dots,x_k$
such that they satisfy the following condition:
$...
2
votes
1
answer
86
views
Pseudo-partitions of $\mathbb{N}$
This question is loosely inspired by the exact cover / partition problem in computer science.
Let $X\neq \emptyset$ be a set and let ${\cal E}\subseteq {\cal P}(X)$. For $x\in X$ we let $c_{\cal E}(x) ...
- 48.8k
0
votes
1
answer
162
views
Formula for partitions of integers with no subpartition being a partition of $t$
When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints ...
- 23
3
votes
4
answers
380
views
Bijections on the set of integer partitions of $n$
I am looking for natural bijections from the set of integer partitions
of $n$ to itself. Of course, I have no definition of natural, but for
the purpose of this question it suffices that it appears ...
- 5,822
1
vote
1
answer
82
views
The sum of the signs of conjugacy classes in the symmetric group S_n [duplicate]
Let $r$ be the number of conjugacy classes of the symmetric group $S_n$ whose sign is $1$, i.e.
\begin{equation}
r := \#\{c \in \text{Conj} (S_n): \text{sgn} (c) = 1 \}.
\end{equation}
Let $s$ be the ...
5
votes
1
answer
212
views
Is the partition tiling relation transitive?
The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but ...
- 48.8k
4
votes
2
answers
307
views
Lower bounding a partition-related sum
We say the $\mathbb{N}$-valued, non-increasing, eventually zero sequence $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$ is a partition of $N$ if $|\lambda|:=\sum_{k\geq 1}\lambda_k=N$, and denote $m_k(\...
- 715
6
votes
0
answers
268
views
A matroid parity exchange property
As part of my research, I encountered the following problem. Let $M = (E,I)$ be a matroid and let $P = \{P_1,\ldots,P_n\}$ be a partition of $E$ into (disjoint) pairs. For $A \subseteq P$, we say that ...
- 173
1
vote
1
answer
472
views
Conjectured upper bound on the maximum value of the absolute value of the Möbius function in the poset of multiplicative partitions under refinement
PRELIMINARIES:
Consider the poset $(\mathcal{P}_n, \leq_r)$ of the (unordered) multiplicative partitions of $n$ partially ordered under refinement (for all $\lambda, \lambda’ \in \mathcal{P}_n$, we ...
- 405
4
votes
1
answer
308
views
3 divides coefficents of this $q$-series
Denote $\phi(q):=\prod_{j\geq1}(1-q^j)$ and let $\xi=e^{\frac{2\pi i}3}$ be a cube root of unity.
Define the sequence $u(n)$ by
$$\prod_{n\geq1}\prod_{s=1}^2(1-q^n\xi^{ns})(1-q^{2n}\xi^{ns})
=\sum_{n\...
- 43.2k
1
vote
0
answers
73
views
Ordered combinatorial classes and partitions
Let $\mathcal{C}$ be a combinatorial class and let $\leq$ be a partial order on $\mathcal{C}$. We say that $(\mathcal{C},\leq)$ is an ordered combinatorial class if for all $x,y\in\mathcal{C}$, $$x&...
- 11
11
votes
0
answers
290
views
Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
7
votes
2
answers
478
views
Upper bound on VC-dimension of partitioned class
Fix $n,k\in \mathbb{N}_+$.
Let $\mathcal{H}$ be a set of functions from $\mathbb{R}^n$ to $\mathbb{R}$ with finite VC-dimension $d\in \mathbb{N}$. Let $\mathcal{H}_k$ denote the set of maps of the ...
- 364
4
votes
1
answer
206
views
Partition numbers as the specific sums of the A161511
Let $p(n)$ be A000041 i.e. number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $a(n)$ be A161511 i.e. number of $1\cdots0$ pairs in the ...
- 4,938
0
votes
0
answers
217
views
On characters of the symmetric group: Part 2
This question is related to my earlier MO quest.
For an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\...
- 43.2k
7
votes
2
answers
439
views
A generalized matroid exchange property
Let $(E,I)$ be a matroid, and let $A,B \in I$ be disjoint independent sets in the matroid. Moreover, let $B_1,\ldots, B_k$ be a partition of $B$. I could not decide if the following is always true. ...
- 173
5
votes
1
answer
374
views
Closed-form for the number of partitions of $n$ avoiding the partition $(4,3,1)$
Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$.
We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrers ...
- 4,938
2
votes
0
answers
352
views
On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
- 43.2k
2
votes
0
answers
125
views
Inequality for 2-associated Stirling numbers of the second kind
Let $S_2(n,k)$ denote the 2-associated Stirling number of the second kind for $n$ objects and $k$ blocks, with $n$ being at least two. That is, we partition $n$ labeled objects into $k$ unlabeled ...
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,938
3
votes
0
answers
222
views
Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
- 405
0
votes
0
answers
170
views
Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter. We ignore the case of $(m_1,\cdots,m_k)=n$.
I am ...
- 405
4
votes
0
answers
211
views
Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
3
votes
2
answers
149
views
Finding an inclusion-based path through 2-part set partitions
Given $S = \{1, 2, \ldots, n\}$, consider partitions of $S$ of the form $(R, R')$ where $R \subset S$ and $R'$ is $S \setminus R$, the complement of $R$ in $S$. The goal is to list 2-part partitions ...
- 4,589
1
vote
0
answers
95
views
Pretty simple recursion for the A290383
Let $a(n)$ be A290383 i.e. number of set partitions of $[n]$ such that the smallest element of each block is odd. Here
$$
a(n)=b(n,0,0)
$$
where
$$
b(n,m,t)=\sum\limits_{j=1}^{m-t+1}b(n-1,\max(m,j),1-...
- 4,938
2
votes
1
answer
222
views
Number of partitions of an integer subject to some restrictions
Given a multiset $S$ of integers and an integer $n$. The size of $S$ is $n$ and each of the elements of $S$ lie within the range $1$ to $n-1$. Give a tight upper bound (in terms of $n$) on the number ...
4
votes
2
answers
343
views
Number of partitions of $n$ and number of different integers in 1-avoiding partitions
Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by
$$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
I have encountered an interesting enumeration.
Take ...
- 43.2k
2
votes
0
answers
72
views
Recursion for the number of partitions of $m^n-1$ into powers of $m$
Let $a(n,m)$ be the number of partitions of $m^n-1$ into powers of $m$. In other words,
$$a(n,m)=[z^{m^n-1}] \prod\limits_{k\geqslant 0} \frac{1}{1-z^{m^k}}$$
Let
$$
R(n,m,q)=\sum\limits_{j=0}^{m(q+1)-...
- 4,938
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 ...
- 51
8
votes
1
answer
334
views
What is the Möbius function for the lattice of partial partitions?
Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of
$\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
- 557
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\...
- 43.2k
0
votes
0
answers
186
views
Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?
For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
- 15.6k
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
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 (...
- 43.2k