# Questions tagged [partitions]

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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### Olympiad problem relevant to $(a,b)$-feasible pair

Recently, a mathematical olympiad problem is proposed as follows: Let $G$ be a graph with $|V| = 100$ and $\delta(G) \geqslant 10$. Prove that there is an integer $0 \leqslant k \leqslant 5$, such ...
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### Intersection numbers of moduli spaces and noncrossing partitions

The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
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### The number of partitions of a positive integer allowing at most r repetitions of any part

Let $q_r(n)$ be the number of partitions of the positive integer $n$ allowing at most $r$ repetitions of any of the parts. (For $r=1$ this is just the usual number of partitions of $n$ into distinct ...
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### Validating a result on evaluating Jack polynomials

I am currently working through the following paper: Lapointe L., Lascoux A., Morse J. Determinantal Expression and Recursion for Jack Polynomials Electron. J. Combin. 7 (2000), Notes 1. DOI: 10.37236/...
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### Fast computation of the partition function modulo a prime

Rademacher’s formula for the partition function allows fast computation using high precision arithmetic, but requiring a lot of memory. Here is an example computation of $p(10^{20})$ by Fredrik ...
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