# Tagged Questions

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

Let $\lambda$ and $\sigma$ be n-partitions $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$ Then let $M_{\lambda \sigma}$ be the number of ways to colour blocks of ...
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### 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 ...
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### Faster formula to compute sum over partitions

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