1
vote
1answer
120 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 ...
1
vote
0answers
118 views

How to prove this identity? (Perhaps related to partition) [closed]

How to prove this identity? $ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}= \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$ I will appreciate it a lot if a solution using method ...
1
vote
1answer
235 views

An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product ...
5
votes
1answer
156 views

The number of partitions between two fixed partitions

Given two partitions M and N, with $M_i \leq N_i$ for all $1\leq i\leq \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum_{\lambda: M_i\leq \lambda_i\leq N_i} ...
6
votes
2answers
140 views

Length of the longest chain in dominance order

If $\Pi_n$ is the set of partitions of $n$, then for $\lambda, \mu\in \Pi_n$ we say $\mu$ dominates $\lambda$ if $\sum\limits_{i=1}^k \lambda_i \leq \sum\limits_{i=1}^k \mu_i$ for all $k$. This gives ...
4
votes
0answers
97 views

Generating random weak k-bounded reverse plane partitions

Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that ...
3
votes
0answers
63 views

Asymptotics of partitions in at most n parts, bounded by r

I posted this question on MathStackexchange (http://math.stackexchange.com/questions/639878/asymptotics-of-partitions-in-at-most-n-parts-bounded-by-r) some time ago, but it did not receive any answer, ...
3
votes
1answer
278 views

Combinatorial Technique Needed

The following problem is likely too special for MO. However I have no clue how to deal with it, so I'll just try. Nevertheless it is a combinatorial problem and a discussion about general methods in ...
4
votes
0answers
165 views

Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property? Definitions: A subset S of the xy-plane is star-convex if there is ...
2
votes
2answers
237 views

Computing the lexicographic indices of integer partition

If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...
2
votes
1answer
274 views

Number of 3-tuple partitions of a multiple of three which follow the triangle inequality

Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ . If I chose any one partition randomly from ...
10
votes
3answers
902 views

A problem on a specific integer partition

Let $n$ be a positive integer, we consider partitions of the following form : $$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that : $d_{i}\vert n$ $1=d_{1}<d_{2} \le d_{3} \le ... \le ...
2
votes
1answer
439 views

Recurrence relation for coefficients of product of generating functions for partition numbers

It is well known that $$Z(x,q) = \prod_{n=1}^\infty\frac{1}{(1-xq^n)} = \sum_{m=0}^\infty\sum_{k=0}^m p_{m,k}x^kq^m$$ is the generating function for the number $p_{m,k}$ of partitions of $m$ in ...
4
votes
2answers
311 views

Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...
3
votes
2answers
138 views

Databases for sequences indexed by partitions

Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...
3
votes
1answer
164 views

bounded partitions and bounded signed partitions of integers

Define a bounded signed partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation: $$n = \pm a_{1} \pm a_{2} \pm a_{3} \pm \dots \pm a_{m}$$ where each $a_{i}$ is a integer in ...
5
votes
2answers
171 views

Random RSK and Plancherel Measure

Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...
3
votes
1answer
111 views

Why are the dinv-statistic and the partition length equidistributed?

A partition of $n$ is a weakly decreasing sequence of natural numbers $\lambda = (\lambda_1, \lambda_2, \dots)$ such that $\sum \lambda_i = n$. Its length $l(\lambda)$ is the number of positive ...
10
votes
0answers
316 views

3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...
5
votes
2answers
247 views

Semimagic Squares and Partitions

Say, we have a semimagic square $X$, that is, an $n\times n$ square matrix with entries from natural numbers, such that each row and column of it sums up to the same natural number $s$. Let $M$ be a ...
6
votes
1answer
296 views

Partitions comprised only of divisors

How many of the partitions of a natural number $n$ are comprised only of its divisors? That is, if $$p(n)=\sum_{\sum_{1}^n kj_k=n:j_k\geq 0} 1_{\[j_1,j_2,...\]},$$ is the ordinary partition function ...
11
votes
3answers
583 views

Partitions into parts from an arithmetic progresion

Fix an arithmetic progression $R=(a, a+m, a+2m, \ldots)$, and assume that $gcd(a,m)=1$. Define $q_R(n)$ as the following coefficients: $$\prod_{i=0}^\infty (1+ t^{a+mi}) = \sum_{n=0}^\infty q_R(n) ...
2
votes
3answers
948 views

Explicit formula for the number of compositions with m strictly positive parts bounded by n?

Is there any known formula for the number of compositions of an integer k (partitions with considering the order of the parts) of length m (exactly m parts) where the parts do not exceed a given ...
5
votes
1answer
192 views

Functions consistent with two partitions of a finite set

Suppose that $P$ and $C$ are two unordered partitions of $[n]$, the set of positive integers from 1 to $n$. Let $c(C,P,x)$ be the number of functions $f$ from $[n]$ to $[x]$ for which (1) ...
5
votes
2answers
391 views

A product identity for partitions

For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let $m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | ...
2
votes
1answer
179 views

Box-dual of a partition - what is it called?

Fix natural numbers $n,m\in\mathbb{N}$. Given a partition $\lambda\vdash d$ with at most $n$ rows (and at most $m$ columns), we can define a partition ...
2
votes
0answers
124 views

Partitions limit shape and LDP

Hello! I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform ...
7
votes
2answers
726 views

Inverse map for partition transform

Let $(a_n)$, $n\in\mathbb{N}$, be a sequence of complex numbers, then formally one has (1) $$\prod_{1}^{\infty}\left(1-a_nx^n\right)^{-1}=1+\sum_{1}^{\infty}\left(\sum_{j_1+2j_2+\cdots ...
8
votes
1answer
175 views

shape of random q-weighted lattice path

Where can I find a detailed write-up of the asymptotic shape of a $q$-weighted Young diagram inside an $a$-by-$b$ box, especially one that uses a variational approach? Equivalently, we can look at ...
7
votes
1answer
284 views

Distribution of big component of set partitions

Consider the set $S_n = \{1, \dotsc, n\},$ and consider the set $P(n, k)$ of partitions of $S_n$ into $k$ parts (the cardinality of $P(n, k)$ is the Stirling number of the second kind $S(n, k).$ ...
2
votes
0answers
276 views

What is 'arch' in Vershik-Kerov's 1984 paper?

In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperator{\arch}{arch}\arch ...
4
votes
2answers
558 views

How local the property of “being a partition” is?

Note: The problem is solved! See EDIT below. The following question about integer partitions arose from a purely "practical" question: Does there exist better dynamic programming algorithms for the ...
4
votes
2answers
3k views

Seeking a solution algorithm to the 3-partition problem

I need to divide 48 pieces of jewelry between 3 inheritors so as to give equal, or nearly equal value, to each. I have learned that this is called the 3-partition problem. I solved it for 9 pieces of ...
4
votes
1answer
274 views

Article about partitions with forbidden parts/multiplicities

I am looking for an article, most likely from the 90s, that generalized the bijection between partitions with odd and distinct parts by explaining how a bijection between the forbidden parts could be ...
7
votes
3answers
746 views

Binomial coefficient in Andrews' partition book

First of all, i think MathOverflow is a very great community to discuss math, either basic or advanced, and i'm glad to participate here. It's my first post, so i'm sorry if i did anything wrong, and ...
5
votes
2answers
813 views

What are the best known bounds on the number of partitions of $n$ into exactly $k$ distinct parts?

For example, if $n = 10$ and $k = 3$, then the legal partitions are $$10 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2$$ so the answer is $4$. By choosing $k$ random elements of $\{1,\ldots,2n/k\}$, ...
1
vote
2answers
307 views

Partitions into 0,1, and 2 with a partial sum condition.

On a tangent to a problem I've been working on, I've run into a combinatorial/partition-theoretic problem that I wondered if anyone had run into before. Let $N$ be a positive integer, and ad-hoc-ly ...
4
votes
1answer
460 views

a simple combinatorial problems

what sequence $C=(c_1,\cdots,c_n)\in \mathbb{N}^n$ with $\sum ic_i=n$ maximizes the number of $\omega\in\mathfrak{S}_n$ of type $C$? For instance,when $n=4$ the maximizing sequence is $(1,0,1,0)$.
1
vote
1answer
480 views

Split sum into equal terms

Given a sum of $l$ integers $r_1+...+r_k+...+r_l$ and an integer $t$. Find indices $1 < p_1 <...< p_h <...< p_{t-1} < l$ such that in sum ...
20
votes
2answers
840 views

Partitions to different parts not exceeding $n$

Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or ...
2
votes
1answer
418 views

Does this type of partition have a name?

If this question is dumb please excuse me. Does this type of partition have a name and if so, what is it? A sequence of partitions of an integer $\vec{\lambda}_1, ...
1
vote
0answers
175 views

Explicit formulas for the action of the Hall algebra of the cyclic quiver on q-Fock space?

In their paper on the decomposition numbers of Schur algebra, Vasserot and Varagnolo introduce an action of the (twisted) Hall algebra of a cyclic quiver $\Gamma$ on q-Fock space. Without q-shifts, ...
5
votes
3answers
548 views

How to recover partition from its multiset of hook lengths?

One of the invariants associated to a partition is its multiset of hook lengths. For instance, as shown here, the partition (5,4,1) has hook lengths {1,1,1,2,3,3,4,5,5,7}. Is there a good way to go ...
23
votes
4answers
1k views

Asymptotic growth of a certain integer sequence

Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence A019568 defined as follows: $a(n):=$ the smallest positive integer $k$ such ...
3
votes
4answers
948 views

unique integer partitions

Let me motivate my general question with an explicit example: Suppose I am looking for all unique combinations of exactly three non-negative integers that sum to five. The solutions are 005, 014, ...
10
votes
1answer
460 views

I am searching for the name of a partition (if it already exists)

I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a ...
1
vote
2answers
431 views

Unique way to partition into two parts of equal weight

A special case says it all ... Let $ w_1 < w_2 < \ldots < w_{12} $ be an increasing sequence of $12$ integers ("weights") such that the total weight $W=\sum_{k=1}^{12}w_k$ is even. Say that ...