# Questions tagged [partitions]

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### 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**

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**2**answers

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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 ...

**5**

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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 ...

**2**

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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

**1**answer

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### 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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**2**answers

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**

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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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### 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 ...

**5**

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### 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 ...

**2**

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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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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**

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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

**1**answer

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 $G$ ...