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Your data matches 249 different statistics following compositions of up to 3 maps.
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Matching statistic: St001276
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St000665
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [3,2,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [1,3,2] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [3,4,1,2] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => [3,4,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => [3,1,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => [4,1,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,4,2,1] => [1,4,3,2] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => [2,4,1,3] => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4,2,3,1] => [2,4,3,1] => 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => [2,1,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [1,2,4,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [3,4,5,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => [3,4,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => [3,4,1,5,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => [4,5,1,2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,4,3,1,2] => [4,5,3,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => [4,1,5,3,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => [4,2,5,1,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => [4,2,5,3,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => [4,2,1,5,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => [1,4,2,5,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => [5,2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => [5,3,4,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => [5,3,1,4,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => [5,4,1,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => [5,4,2,3,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => [5,4,3,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => [1,5,4,3,2] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => [2,5,4,1,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [5,3,4,2,1] => [2,5,4,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => [2,1,5,4,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => [1,2,5,4,3] => 1
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Matching statistic: St001465
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [1,4,3,2] => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,2,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,4,5,3,1] => [1,2,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,5,2,4,1] => [2,5,1,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [1,3,2,5,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,2,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => [1,3,2,5,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,5,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,5,3,2] => [1,2,5,4,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,2,5,4,3] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,4,5,1,3] => [1,3,5,2,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [1,4,3,2,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [1,3,5,2,4] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [1,4,3,2,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [2,1,4,3,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,2,4,3,5] => 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St001086
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,2,4] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [1,4,5,2,3,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,6,4,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [1,6,4,5,2,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [1,4,6,2,3,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => [1,3,2,5,6,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => [1,4,5,6,2,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => [1,4,2,3,6,5] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => [1,5,6,2,3,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => [1,6,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => [1,5,3,4,2,6] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,3,5,2,4,6] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => [1,6,3,4,2,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => [1,6,3,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => [1,3,6,2,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => [1,5,6,3,4,2] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => [1,3,5,6,2,4] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => [1,3,4,2,6,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => [1,2,5,6,3,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => [1,3,4,2,5,6] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,7,4,6] => [1,6,7,4,5,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,2,6,4,7] => [1,6,4,5,2,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,5,2,6,7,4] => [1,7,4,5,2,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,5,6,2,7,4] => [1,7,4,6,2,3,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => [1,5,7,4,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5,7] => [1,5,6,2,3,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,4,6,2,7,5] => [1,7,5,6,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,4,2,5,7,3,6] => [1,6,7,3,4,2,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,2,7,3,6] => [1,6,7,3,5,2,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,2,5,6,3,7] => [1,6,3,4,2,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,5,2,6,3,7] => [1,6,3,5,2,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,5,6,7,3] => [1,7,3,4,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,5,2,6,7,3] => [1,7,3,5,2,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,5,6,2,7,3] => [1,7,3,6,2,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,6,3,7,5] => [1,7,5,6,3,4,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,7,3,5] => [1,5,7,3,4,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,7,3,5] => [1,5,7,3,6,2,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5,7] => [1,5,6,3,4,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,2,7,3,5,6] => [1,5,6,7,3,4,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,3,7,4] => [1,7,4,6,3,5,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,5,2,6,7,3,4] => [1,4,7,3,5,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,5,6,2,7,3,4] => [1,4,7,3,6,2,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,5,2,7,3,4,6] => [1,4,6,7,3,5,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001222
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,3}
Description
Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.
Matching statistic: St001606
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 79%●distinct values known / distinct values provided: 75%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 79%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
Matching statistic: St001629
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => [1] => ? = 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1] => ? ∊ {0,1}
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => [1] => ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2] => ? ∊ {0,1,1,1}
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1] => ? ∊ {0,1,1,1}
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1] => ? ∊ {0,1,1,1}
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1] => ? ∊ {0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [1,2] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [1,2] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => [1,2] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1] => ? ∊ {1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => ? ∊ {1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [1,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,2] => [1,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => [1,1,1,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,2,1] => [1,1,1,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => [1,1,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => [2,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => [3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,2,1,1] => [2,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,2,1,1] => [2,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => [3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => [1,1,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => [1,1,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => [1,1,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => [1,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => [1,1,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,1] => [1,1,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,1] => [1,1,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => [2] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => [1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => [1,1,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,1,1,1] => [1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,1,1,1] => [1,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => [1,1,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,2,1] => [1,1,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,2,1] => [1,1,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => [1,1,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3] => [2] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,1,1] => [1,2] => 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => [1] => ? ∊ {0,0,0,0,0,0,1,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1] => [1,5] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,1,1,1,2] => [1,3,1] => 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,1,2,1] => [1,2,1,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,1,3] => [1,2,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,1,2,1,1] => [1,1,1,2] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,1,2,2] => [1,1,2] => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,1,2,1,1] => [1,1,1,2] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,1,2,1,1] => [1,1,1,2] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,1,2,2] => [1,1,2] => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,1,3,1] => [1,1,1,1] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,5] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,2] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,2] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,2] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,2] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => [1] => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001714
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St001123
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> [1]
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,0,0,1,0]
=> [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [5,5]
=> [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,3,3]
=> [3,3]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [7,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [5,3]
=> [3]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [7]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [5,2,1,1]
=> [2,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [6,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,3,1]
=> [3,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [6,3]
=> [3]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [5,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [6,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [4,3,1]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,3]
=> [3]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [4,3]
=> [3]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [4,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [5,5,5]
=> [5,5]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [7,7]
=> [7]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,3,3,2]
=> [3,3,3,2]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [5,5,3]
=> [5,3]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,3,3]
=> [3,3,3]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [8,3,3]
=> [3,3]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [9,4]
=> [4]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,3,2]
=> [3,3,2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [7,5]
=> [5]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [5,3,3]
=> [3,3]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [7,3,2]
=> [3,2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [9,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [7,3]
=> [3]
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [9]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [7,4,1,1]
=> [4,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [3,3,2,1,1,1]
=> [3,2,1,1,1]
=> 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [7,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [8,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [8]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [7]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [6,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [6]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [2,1]
=> [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [2]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1]
=> []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St001283
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [2]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,1,1]
=> [3,1,1]
=> 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
The following 239 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000137The Grundy value of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000010The length of the partition. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000306The bounce count of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000475The number of parts equal to 1 in a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000549The number of odd partial sums of an integer partition. St000658The number of rises of length 2 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000783The side length of the largest staircase partition fitting into a partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000897The number of different multiplicities of parts of an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000992The alternating sum of the parts of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001484The number of singletons of an integer partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001939The number of parts that are equal to their multiplicity in the integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000260The radius of a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001114The number of odd descents of a permutation. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001593This is the number of standard Young tableaux of the given shifted shape. St001651The Frankl number of a lattice. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000546The number of global descents of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000779The tier of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St001728The number of invisible descents of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000486The number of cycles of length at least 3 of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001961The sum of the greatest common divisors of all pairs of parts. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001597The Frobenius rank of a skew partition. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000478Another weight of a partition according to Alladi. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000993The multiplicity of the largest part of an integer partition. St001383The BG-rank of an integer partition. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the 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. 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. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000454The largest eigenvalue of a graph if it is integral. St001890The maximum magnitude of the Möbius function of a poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000366The number of double descents of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000741The Colin de Verdière graph invariant. St000007The number of saliances of the permutation. St001895The oddness of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001096The size of the overlap set of a permutation. St000731The number of double exceedences of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001330The hat guessing number of a graph. St000891The number of distinct diagonal sums of a permutation matrix. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000365The number of double ascents of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000650The number of 3-rises of a permutation. St000732The number of double deficiencies of a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001513The number of nested exceedences of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000654The first descent of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001202Call 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. St001481The minimal height of a peak of a Dyck path. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000061The number of nodes on the left branch of a binary tree. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000850The number of 1/2-balanced pairs in a poset. St001399The distinguishing number of a poset. St001624The breadth of a lattice. 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 $eA$.
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