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      Questions tagged [partitions]

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      5
      votes
      1answer
      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 $\eta(k):=\sum_jk_j$. ...
      2
      votes
      2answers
      139 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
      votes
      0answers
      79 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
      117 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
      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{...
      6
      votes
      1answer
      261 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 \...
      16
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      2answers
      998 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 $\...
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      55 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
      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\...
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      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 ...
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      votes
      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 ...
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      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(\...
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      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 ...
      1
      vote
      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 ...
      1
      vote
      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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