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# Questions tagged [partitions]

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Let $\Phi_+$ be the set of positive roots in some root system, and let $Q_+$ be the positive part of the root lattice, i.e., the set of elements of the form $\sum_{\beta\in \Phi_+}m_\beta\beta$ with $... 0answers 87 views ### Complementary Bell numbers$B^{\pm}(24n+14)$The complementary Bell numbers$B^{\pm}(n)$are defined by the alternating sum of the Stirling numbers of the second kind,$S(n,k)$: $$B^{\pm}(n)=\sum_{k=0}^n(-1)^kS(n,k),$$ and they count the ... 4answers 369 views ### Maximum conjugacy class size in$S_n$with fixed number of cycles Context: It is well known that given a permutation in$S_n$with$a_ii\$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by $$\frac{n!}{\prod_{j=1}^... 1answer 109 views ### Collapsed partitions and ordinary partitions Adopt the standard notation for integer partitions, writing \lambda_1^{a_1} \cdots \lambda_k^{a_k} as shorthand for the partition a_1 \lambda_1 + \cdots + a_k \lambda_k with parts \lambda_1 > \... 2answers 275 views ### Number of integers for which np(n) is a perfect square Let p(n) be the partition function. Are n=1,2,3 the only cases for which np(n) is a perfect square? 1answer 108 views ### Reading off top hook-lengths in partitions Given an integer partition \lambda and its Young diagram Y_{\lambda}, let h_{\lambda}(i,j) stand for the corresponding hook length of the cell (i,j)\in Y_{\lambda}. Write \lambda\vdash n for ... 1answer 351 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. ... 2answers 146 views ### 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 ... 0answers 83 views ### 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, ... 0answers 120 views ### 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 ... 0answers 106 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{... 1answer 269 views ### hook-length formula: “Fibonaccized”: Part II This is a natural follow-up to my previous MO question, which I share with Brian Hopkins. Consider the Young diagram of a partition \lambda = (\lambda_1,\ldots,\lambda_k). For a square (i,j) \in \... 2answers 1k views ### hook-length formula: “Fibonaccized” Part I Consider the Young diagram of a partition \lambda = (\lambda_1,\ldots,\lambda_k). For a square (i,j) \in \lambda, define the hook numbers h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1 where \... 0answers 58 views ### Is the fundamental partition associated to n the partition of r_{0}(n) in k_{0}(n) parts that maximizes entropy? 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 334 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\...

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