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      Questions tagged [partitions]

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      1answer
      57 views

      Algorithm to list all Kostant partitions

      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 $...
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      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 ...
      7
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      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 $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by $$ \frac{n!}{\prod_{j=1}^...
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      votes
      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 > \...
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      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?
      2
      votes
      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 ...
      5
      votes
      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 $\eta(k):=\sum_jk_j$. ...
      2
      votes
      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 ...
      5
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      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, ...
      3
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      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 ...
      2
      votes
      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{...
      6
      votes
      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 \...
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      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 $\...
      0
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      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-...
      5
      votes
      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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