The partitions tag has no usage guidance.

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### An inequality on partitions into distinct bounded parts

Let $P(n,m)$ denote the set of all positive integer partitions of $n$ into parts that are pairwise distinct and bounded by $m$. Let $p(n,m) = |P(n,m)|$.
After some numerical experiments it appears
$...

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**0**answers

172 views

### Enumerating a class of polynomials

How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...

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**1**answer

250 views

### Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one
which for him represented the best of Ramanujan. There are many proofs for this identity.
(for example, H. H. Chan’s proof, M. Hirschhorn's proof....

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**1**answer

106 views

### Algorithm for finding numbers with an even partition number

NOTE: After edit question became about set partitions, which not was I intended, so this is second try.
Is there an algorithm for producing an infinite subset of set of integer partition numbers p(n) ...

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**2**answers

219 views

### sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms.
Thanks,

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86 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 ...

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**1**answer

96 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 ...

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22 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 ...

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165 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 (x)_{k_{r-1}-k_r}(x)_{k_r}}=\...

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**2**answers

552 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:
$$\dfrac{...

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99 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 $[...

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**1**answer

117 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 $...

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54 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 ...

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**1**answer

82 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 :=\...

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**1**answer

191 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 known/...

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420 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 $n$ ...

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321 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 ...

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164 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 ...

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**0**answers

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### 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 ...

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**1**answer

271 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 ...

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### 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 $\...

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64 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 ...

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**1**answer

173 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 ...

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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 ...

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78 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 \frac{1}{(1-q^{8n-3})(1-...

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### 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 ...

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### 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 ...

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201 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 ...

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67 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 (q)_{N}}$...

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324 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 ...

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128 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 ...

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**1**answer

246 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 ...

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### 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 $R(N,L)$ ...

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**1**answer

273 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 ...

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117 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 ...

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**1**answer

160 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 ...

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272 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 ...

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**1**answer

161 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 ...

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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 $(1^{\alpha_1},2^{\alpha_2},3^{\alpha_3},...,s^{\...

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### 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 $\...

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### 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 $$M_f(\...

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134 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 $\text{Part}(...

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### 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 ...

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151 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]} = \{...

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**1**answer

181 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 welcome....

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229 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 ...

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### 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 $\partial K$) ...

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**1**answer

80 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 ...

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### 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 + ...

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### 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
\...