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2
votes
0answers
80 views

Balanced partitions of vector sets

We are interested in the following Lemma. Let $V\subset [0,1]^n\subset\mathbb R^n$ be a set of $n$-dimensional vectors. Then for each $r\le |V|$ there exists a partition $$V=V_1\cup V_2\cup\dots \cup ...
3
votes
1answer
89 views

Partitions into parts differing by 2

If we look at the difference between the number of partitions of $n$ with distinct parts that have an even number of parts and the number of partitions of $n$ with distinct parts that have an odd ...
1
vote
0answers
20 views

Frobenius form of the quotient of a partition

Let $\mu$ be a partition of $n \cdot l$ with a trivial $l$-core. To $\mu$ we can associate an $l$-multipartition of $n$ called the $l$-quotient of $\mu$. There are several equivalent descriptions of ...
3
votes
0answers
162 views

A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form $$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...
5
votes
2answers
548 views

mod 5 partition identity proof

I am looking for a proof that: $$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$ The left hand side expands into: ...
1
vote
0answers
94 views

The lattice of graphs under vertex abstractions

I am curious to know if the following structure has been studied, or if anything similar is in the literature. For $n \in \mathbb{N}$, let $G = ([n],E)$ be a digraph. A partition of a subset $V$ of ...
-5
votes
1answer
111 views

Automorphisms of partitions [closed]

I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in ...
1
vote
0answers
51 views

Rank-unimodality and Sperner property of higher dimensional partitions

I did a google-search but have not been able to find much reference on this problem. So I am asking it here hoping to get some information. Consider the set of all 4-dimensional Ferrer's diagram ...
2
votes
1answer
78 views

Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha ...
3
votes
1answer
180 views

Restricted integer partitions modulo k

Let $p(n,m)$ be the number of partitions of the integer $n$ into exactly $m$ parts. Consider the sequence $a_n = p(n,m)$. What is known about the sequence $a_n$ mod $k$? In particular, is it ...
11
votes
1answer
413 views

Generating function for certain partitions (with a restriction on the Durfee square)

First of all my apologies if this question is well known or obvious: this is not in my area of research. Let $T(x)=\sum_{n=0}^\infty t_nx^n$, where $t_n$ is the number of partitions $\lambda$ of $n$ ...
5
votes
4answers
310 views

Counting refinements of partitions

Let $p$ and $q$ be partitions of $n$. We say $q$ refines $p$ if the parts of $p$ can be subdivided to produce the parts of $q$. For example, $(5,5,1)$ refines $(6,5)$ but not $(7,4)$. $(n)$ refines ...
1
vote
1answer
159 views

The number of good partitions

This was also posted in stackexchange. However, I have no idea how difficult it is. All hints or references are appreciated! Consider a set $S$ of $n$ red balls and $m$ blue balls. It is well known ...
2
votes
0answers
89 views

Number of multipartite partitions with odd components

For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all ...
5
votes
1answer
259 views

A remarkable sum over partitions

While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to ...
3
votes
0answers
125 views

How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$

Let $\lambda$ be a partition of an positive integer $n$, it can be presented as $\lambda=(\lambda_{1},\lambda_2,\cdots,\lambda_l)$ such that $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_l>0$, or ...
0
votes
0answers
63 views

Factorial Sums over Compositions or ``Unlabeled Permutations"

Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$ In a divergent sum, the sequence $$ a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i! $$ frequently shows up and one ...
4
votes
1answer
166 views

Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition ...
3
votes
0answers
41 views

Root Polylogarithm Dominance Questions

Motivation: I am trying to work on a problem related to computing the roots of a certain family of polynomials related to integer partition theory. In particular, I have been trying to ``Bridge the ...
7
votes
0answers
73 views

A conjectured q-continued fraction related to the Göllnitz-Gordon partition identities

Given $q=e^{2i\pi\tau}$ with $|q|\lt1$, define the well-known Göllnitz-Gordon identities $$A(q)=\sum_{n=0}^\infty \frac{q^{n(n+1)}(-q;q^2)_n}{(q^2;q^2)_n}=\prod_{n=1}^\infty ...
3
votes
0answers
64 views

What is the maximal number of partitions with this maximal intersection property?

Let $X = \{ 1, \dots, n = sk \}$ be a finite set. Let $\mathscr P, \mathscr Q$ be equi-partitions of $X$ into $k$ sets of size $s$. Denote by $V(\mathscr P, \mathscr Q)$ the maximum size of ...
3
votes
1answer
127 views

Complexity of a very simple graph partitioning problem

The following problem seems like a very simple and natural one, but I am not familiar with any existing work on it; in particular I am hoping to prove it is NP hard: Let $G$ be a complete weighted ...
-4
votes
1answer
198 views

patitions of the number n [closed]

I'm having difficult with the following question : A. Show that the number of partitions of n where in each one of them the even numbers appears at most once equals to the number of partitions of n ...
1
vote
1answer
64 views

Restricted partitions with square terms only

Let $P(N,M,n)$ be the number of partitions of $n$ such that each term is $\le N$ and there are at most $M$ terms. So we know the generating function for $P(N,M,n)$ is $ \frac{(q)_{N+M}}{(q)_M ...
5
votes
1answer
323 views

Terminology in combinatorics

I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics. A finite graph $G$ has the following ...
1
vote
0answers
120 views

Randomly partitioning the unit interval with continuous functions

I want to construct a family of continuous functions $H$ in order to randomly partition the unit interval. That is, consider a partition $\lambda$ of the unit interval into $n$ subintervals: $\lambda ...
5
votes
1answer
221 views

Set counting problem with a cap on the intersection between the set and a fixed partition

Fix sets $T_1,\ldots T_m$ as a $k$-partition of $[m\cdot k]=\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. 1) For any $j\le k$, how many sets $C\subset [m\cdot k]$ are there ...
0
votes
0answers
90 views

Classic question on integer partitions (with distinct summands)

I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem: Denote $R(N,L)$ ...
4
votes
1answer
268 views

Euler-like identity for partition function

Following is the wonderful Euler's partition identity: $$\prod_{i=1}^\infty (1 - x^i) = 1 + \sum_{k=1}^\infty (-1)^k \left (x^{(3k^2-k)/2} + x^{(3k^2+k)/2} \right )$$ I'm wondering if there is ...
1
vote
0answers
113 views

How should this multinomial identity be written?

Question if it is correct, how is the identity tagged (4) below usually written, and can the use of conjugate partitions be avoided? Motivation I apologize for the length of this question - it's as ...
5
votes
1answer
159 views

A generalization of Erdős–Newman–Mirsky?

Given a sequence $S$ of natural numbers, write ${\bf Gap}(S)$ for the set of differences between consecutive terms. (So $|{\bf Gap}(S)|=1$ precisely for arithmetic progressions, hence the connection ...
19
votes
0answers
269 views

Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ...
3
votes
1answer
157 views

A question of terminology regarding integer partitions

I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_r\geq 1$ of $n$ into $r$ parts. Then we can ...
-2
votes
1answer
122 views

Combinatorical meaning of such expression [closed]

Any combinatorical meaning or interpretation of $$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$ for partition ...
3
votes
0answers
109 views

Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...
3
votes
0answers
80 views

Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix ...
2
votes
1answer
132 views

Infimum of partitions

Let $X\neq\emptyset$ be a set. A partition is a subset $P\subseteq {\cal P}(X)\setminus \{\emptyset\}$ such that $\bigcup P = X$ and any distinct members of $P$ are disjoint. We denote by ...
1
vote
0answers
94 views

Optimal tiling for a collection of partitions

I'm interested in a possible generalization of Tiling relation on the set of partitions (the question has only been partially answered). Let $x$ be an infinite set and let $\text{Part}(x)$ be the ...
2
votes
1answer
148 views

Tiling relation on the set of partitions

Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$ and $t\subseteq x$. We set $$P_{[t]} = ...
1
vote
1answer
167 views

asymptotic for restricted partitions

Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers. Is there an asymptotic formula for $P(n,m)$ ?? Any reference is ...
8
votes
1answer
228 views

For any two noncrossing partitions $p, q$ of $n$, is the graph of geodesics from $p$ to $q$ in $NC(n)$ connected?

Let $NC(n)$ denote the lattice of noncrossing partitions of $n$, and let $G$ denote the Hasse diagram of $NC(n)$ with respect to covering relations, viewed as an undirected graph. I'm interested in ...
6
votes
0answers
86 views

Unbalanced equipartitions

Let $K$ be a compact convex set in the plane. Say that a perimeter-halving partition of $K$ is a partition of $K$ into two pieces by a chord (a segment with endpoints on the boundary $\partial K$) ...
3
votes
1answer
77 views

Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...
20
votes
1answer
1k views

Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
7
votes
1answer
195 views

Generating function of $p(25n + 24)$

In a paper titled "RAMANUJAN’S UNPUBLISHED MANUSCRIPT ON THE PARTITION AND TAU FUNCTIONS WITH PROOFS AND COMMENTARY" by Bruce C. Berndt and Ken Ono, it is mentioned that Ramanujan derived the formula ...
6
votes
1answer
186 views

Coloring summands of given n-partition with given weights of colors

Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$ Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...
6
votes
1answer
280 views

A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless of the starting $n$. For example, let $n=69$, and consider this partition: $$ (8,8,7,7,5,5,5,5,5,4,3,3,2,2) $$ In ...
9
votes
2answers
313 views

Faster formula to compute sum over partitions

Let $f$ be a function from the positive integers to the real numbers (or some ring...). Let $$(\star) \quad F(n) = \sum_{n_1 \leq \cdots \leq n_j\atop n_1 + \cdots + n_j = n} f(n_1) \cdots f(n_j), $$ ...
4
votes
0answers
188 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
1
vote
1answer
145 views

Sequences that represent different drawing of chords?

In combinatorics there are there are special kind of sequences, in which their terms represent the number of different ways that we can draw chords with some properties. Actually my question is ...