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Matching statistic: St001491
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Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1]
=> 10 => 1
[2,1]
=> [1]
=> [1]
=> 10 => 1
[1,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[3,1]
=> [1]
=> [1]
=> 10 => 1
[2,2]
=> [2]
=> [1,1]
=> 110 => 1
[2,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[4,1]
=> [1]
=> [1]
=> 10 => 1
[3,2]
=> [2]
=> [1,1]
=> 110 => 1
[3,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[2,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[5,1]
=> [1]
=> [1]
=> 10 => 1
[4,2]
=> [2]
=> [1,1]
=> 110 => 1
[4,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[3,3]
=> [3]
=> [1,1,1]
=> 1110 => 2
[3,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1
[6,1]
=> [1]
=> [1]
=> 10 => 1
[5,2]
=> [2]
=> [1,1]
=> 110 => 1
[5,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[4,3]
=> [3]
=> [1,1,1]
=> 1110 => 2
[4,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1
[7,1]
=> [1]
=> [1]
=> 10 => 1
[6,2]
=> [2]
=> [1,1]
=> 110 => 1
[6,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[5,3]
=> [3]
=> [1,1,1]
=> 1110 => 2
[5,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1
[8,1]
=> [1]
=> [1]
=> 10 => 1
[7,2]
=> [2]
=> [1,1]
=> 110 => 1
[7,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[6,3]
=> [3]
=> [1,1,1]
=> 1110 => 2
[6,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[6,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[5,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1
[9,1]
=> [1]
=> [1]
=> 10 => 1
[8,2]
=> [2]
=> [1,1]
=> 110 => 1
[8,1,1]
=> [1,1]
=> [2]
=> 100 => 1
[7,3]
=> [3]
=> [1,1,1]
=> 1110 => 2
[7,2,1]
=> [2,1]
=> [3]
=> 1000 => 1
[7,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1010 => 0
[6,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1
[10,1]
=> [1]
=> [1]
=> 10 => 1
[9,2]
=> [2]
=> [1,1]
=> 110 => 1
[9,1,1]
=> [1,1]
=> [2]
=> 100 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001913
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1]
=> [1]
=> 1
[2,1]
=> [1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[3,1]
=> [1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[4,1]
=> [1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[5,1]
=> [1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[3,3]
=> [3]
=> [2,1]
=> [2,1]
=> 2
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 1
[6,1]
=> [1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[4,3]
=> [3]
=> [2,1]
=> [2,1]
=> 2
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 1
[7,1]
=> [1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[5,3]
=> [3]
=> [2,1]
=> [2,1]
=> 2
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 1
[8,1]
=> [1]
=> [1]
=> [1]
=> 1
[7,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[6,3]
=> [3]
=> [2,1]
=> [2,1]
=> 2
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 1
[9,1]
=> [1]
=> [1]
=> [1]
=> 1
[8,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[7,3]
=> [3]
=> [2,1]
=> [2,1]
=> 2
[7,2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 1
[10,1]
=> [1]
=> [1]
=> [1]
=> 1
[9,2]
=> [2]
=> [1,1]
=> [2]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> [1,1]
=> 1
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
Matching statistic: St001722
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 60%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 60%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[3,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[7,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[8,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[9,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[7,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[7,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[10,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[8,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[8,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[11,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[9,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[9,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[8,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[12,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[10,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[10,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[9,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[13,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[11,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[11,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[10,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[14,1]
=> [1]
=> [1,0,1,0]
=> 1010 => 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1
[12,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[12,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[11,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[13,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[13,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[12,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[14,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2
[14,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0
[13,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000460
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001247
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
Sum of the odd parts of a partition.
Matching statistic: St001250
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001360
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001360: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001360: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The number of covering relations in Young's lattice below a partition.
Matching statistic: St001378
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001378: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001378: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 67%
Values
[1,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> ? = 1
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,1]
=> [1]
=> [1]
=> []
=> ? = 1
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> ? = 1
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> ? = 1
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[6,1]
=> [1]
=> [1]
=> []
=> ? = 1
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[7,1]
=> [1]
=> [1]
=> []
=> ? = 1
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[8,1]
=> [1]
=> [1]
=> []
=> ? = 1
[7,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[7,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[6,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[9,1]
=> [1]
=> [1]
=> []
=> ? = 1
[8,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[8,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[7,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[6,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[10,1]
=> [1]
=> [1]
=> []
=> ? = 1
[9,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[9,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[8,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[8,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[8,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[7,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[11,1]
=> [1]
=> [1]
=> []
=> ? = 1
[10,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[10,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[9,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[9,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[9,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[8,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[12,1]
=> [1]
=> [1]
=> []
=> ? = 1
[11,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[11,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[10,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[10,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[10,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[9,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[13,1]
=> [1]
=> [1]
=> []
=> ? = 1
[12,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[12,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[11,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[11,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[11,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[10,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[14,1]
=> [1]
=> [1]
=> []
=> ? = 1
[13,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[13,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[12,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[12,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[12,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[11,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[15,1]
=> [1]
=> [1]
=> []
=> ? = 1
[14,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[14,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[13,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[13,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[13,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[12,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[16,1]
=> [1]
=> [1]
=> []
=> ? = 1
[15,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[15,1,1]
=> [1,1]
=> [2]
=> []
=> ? = 1
[14,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[14,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[14,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
Description
The product of the cohook lengths of the integer partition.
For a cell $c = (i,j)$, the '''cohook length''' of $c$ is $h^*(c) = i+j-1$. This statistic is then
$$\prod_{c \in \lambda} h^*(c).$$
The following 175 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000782The indicator function of whether a given perfect matching is an L & P matching. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000635The number of strictly order preserving maps of a poset into itself. St001964The interval resolution global dimension of a poset. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000454The largest eigenvalue of a graph if it is integral. St000627The exponent of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001520The number of strict 3-descents. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000297The number of leading ones in a binary word. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000628The balance of a binary word. St000629The defect of a binary word. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000807The sum of the heights of the valleys of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St000943The number of spots the most unlucky car had to go further in a parking function. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001569The maximal modular displacement of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001424The number of distinct squares in a binary word. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice.
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