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# Questions tagged [partitions]

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230 questions
342 views

### A binary hook-length formula?

This is purely exploratory and inspired by curiosity. Setup: For an integer $k>0$, let $k=\sum_{j\geq0}k_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum_jk_j$. ...
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### Families of ordered set partitions with disjoint blocks

Let $C_1,\dots, C_m$ be a family of ordered set partitions of $[n]$ with exactly $k$ blocks. Write $C_i = \{B_{i1}, \dots, B_{ik}\}$ for $i=1,\dots, m$ where $B_{ij}$ are the blocks of the ordered ...
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### Hooks, monomers, dimers and Young diagrams: Part II

As promised, I've upgraded my last question. Consider the $k$-by-$n$ partition $\lambda_n=(n,\dots,n)$ and its corresponding Young diagram $Y_{n,k}$, which is a $k\times n$ rectangle of cells. Now, ...
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### Hooks, monomers, dimers and Young diagrams: Part I

Following Richard Stanley's pointers regarding my earlier MO question, I decided to "scale-down" the problem and add a slight "twist" to it. Consider the one-line partition $\lambda_n=(n)$ and its ...
97 views

### Newman's conjecture of Partition function

(Sorry for my poor english....) Let $p(n)$ be a partition function and $M$ be an integer. Newman conjectured that for each $0\leq r\leq M-1$, there are infinitely many integers $n$ such that \begin{...
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As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-... 0answers 331 views ###$\text{Determinant}=(\sum \text{Determinant})^2$Denote by$\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$the staircase partition and the embedded partition$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$. QUESTION 1. Is this true? $$\det\... 0answers 127 views ### Formula for number of edges in Hasse diagram of Young's lattice interval There is a determinantal formula for the number of elements of the interval [\mu,\lambda] of Young's lattice between two partitions due to Kreweras and MacMahon in the case of \mu=\varnothing (see ... 0answers 111 views ### Identity for classes of plane partitions There are several classes of plane partitions in the literature. Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ... 0answers 81 views ### How to prove this identity on summations and partitions? Let f be a symmetric function of s variables. The identity is$$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\... 0answers 185 views ### Parity of number of partitions of$n!/6$and$n!/2$The parities of the number of partitions of$n!/6$and$n!/2$appear to be non-random initially, as follows — is there an explanation for this other than chance? With$p$being the partition ... 0answers 66 views ### Shuffling unordered partitions Consider the following: Let$\mathcal{A}$be an unordered partition of$\{1,\dotsc,p\}$, Let$\mathcal{B}$be an unordered partition of$\{1,\dotsc,q\}$Let$\mathcal{C}$be an unordered partition of ... 1answer 70 views ### Enumerating isomorphic subgraphs For digraphs$G$and$H$if we can partition$V(G)$into a family$\{Q_t\}_{t\in V(H)}$indexed by$V(H)$such that$E(G)=\bigcup_{(u,v)\in E(H)}Q_u\times Q_v$, then is every subgraph of$G\$ ...

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