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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St001217
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St001204
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St000654
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1 + 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,3,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,4,3,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,4,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [2,4,1,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,4,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 2 = 1 + 1
[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] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,5,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [3,5,4,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [4,3,5,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [3,5,4,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,4,5,3,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,5,2,4,1] => [3,5,2,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [3,5,2,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [3,2,5,4,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,5,4,3,1] => 2 = 1 + 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] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,5,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,4,5,2] => [1,5,4,3,2] => 2 = 1 + 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] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,4,5,1,3] => [2,5,4,1,3] => 2 = 1 + 1
[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] => 2 = 1 + 1
[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 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [3,5,2,1,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [3,2,5,1,4] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => 2 = 1 + 1
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000781
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001199
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 71%●distinct values known / distinct values provided: 50%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 71%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? = 0
[1,1,0,0]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0}
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [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}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => [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}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => [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}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => [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}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => [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}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => [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}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => [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}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,4,1,6,3,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6,3,1,5,4,2] => [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}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [6,4,1,5,3,2] => [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}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => [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}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [6,3,5,1,4,2] => [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}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [6,5,4,1,3,2] => [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}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6,3,5,4,1,2] => [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}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,7,5,4,3,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,7,6,4,3,2] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,7,6,4,3,2] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [6,1,7,5,4,3,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,7,6,5,3,2] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [7,3,1,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [7,4,1,6,5,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [7,6,1,5,4,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [7,4,1,6,5,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [7,3,1,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [7,4,1,6,5,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [7,3,1,6,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [7,4,1,6,5,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [7,3,6,1,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [7,3,6,1,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [7,3,6,1,5,4,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000284
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000698
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000704
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000706
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? ∊ {0,1}
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,0,1,0,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [2]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [2]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,2]
=> [2]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
Description
The product of the factorials of the multiplicities of an integer partition.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000326The position of the first one in a binary word after appending a 1 at the end. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001933The largest multiplicity of a part in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000864The number of circled entries of the shifted recording tableau of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000659The number of rises of length at least 2 of a Dyck path. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001737The number of descents of type 2 in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000456The monochromatic index of a connected graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001948The number of augmented double ascents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000929The constant term of the character polynomial of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000137The Grundy value of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. 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. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. 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. St001587Half of the largest even part of an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. 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. 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. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a 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. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
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