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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000352
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [2,1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St001113
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St001113: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 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 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[]
=> ? = 0 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St001181
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [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 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [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 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [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 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [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 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [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]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [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 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [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 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,6,3,1,2,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,1,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,7,1,2,3,4,5,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [7,8,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[]
=> []
=> [] => []
=> ? = 0 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 4 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 3 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001186
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 67%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [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 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [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 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [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 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [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 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [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]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [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 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [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]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [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 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [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]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,1,2,7] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,6,3,1,2,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,1,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,7,1,2,3,4,5,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [7,8,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[]
=> []
=> [] => []
=> ? = 0 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 4 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 3 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [7,8,9,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001491
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1] => [1] => => ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => 0 => ? = 1
[1,1,0,0]
=> [1,2] => [1,2] => 1 => 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => 00 => ? = 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 01 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 10 => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 00 => ? = 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,4,3,1] => 000 => ? = 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,4,1,3] => 000 => ? = 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 000 => ? = 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,4,3] => 010 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,4,3,2] => 100 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,4,2] => 000 => ? = 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 000 => ? = 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 000 => ? = 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,4,3,2] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 000 => ? = 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0000 => ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 0000 => ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 0000 => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 0000 => ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 0000 => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 0100 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 0000 => ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,5,4,1,3] => 0000 => ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,5,1,4,3] => 0000 => ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 0100 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 0100 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 0000 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 0100 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,3,2] => 1000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 0000 => ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,5,4,2] => 0000 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,5,1,4,2] => 0000 => ? = 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,5,4,1,2] => 0000 => ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 0000 => ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 0000 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 0000 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => 0000 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 0000 => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => 1000 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 0000 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,5,3,2] => 0000 => ? = 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,3,2] => 0000 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => 0000 => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 1000 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => 1000 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 1000 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0000 => ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 00000 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 00000 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [3,6,5,4,1,2] => 00000 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => 00000 => ? = 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,6,1,5,3,2] => 00000 => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,1,6,5,3,2] => 00000 => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => 00000 => ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [1,6,5,4,3,2] => 10000 => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => [1,6,5,4,3,2] => 10000 => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => 00000 => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => [5,1,6,4,3,2] => 00000 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [1,6,5,4,3,2] => 10000 => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => [1,6,5,4,3,2] => 10000 => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => [1,6,5,4,3,2] => 10000 => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 00000 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => 10000 => ? = 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001095
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2 - 1
[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]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? = 1 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
Description
The number of non-isomorphic posets with precisely one further covering relation.
Matching statistic: St001207
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> []
=> []
=> [1] => ? = 1 + 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,0,0]
=> []
=> []
=> [1] => ? = 1 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> []
=> []
=> [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001198
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 1 + 1
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000237
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 40%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,4,2,3] => 0 = 1 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,5,6,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,2,6,3,5,4] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,4,2,3,6,5,8,7] => ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [1,4,2,3,7,8,5,6] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [1,5,6,2,3,8,4,7] => ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [1,5,7,2,3,4,8,6] => ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [1,6,2,3,4,5,8,7] => ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [1,8,2,6,3,5,4,7] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,2,7,3,5,4,8,6] => ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [1,2,7,8,3,6,4,5] => ? = 1 - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,2,8,3,4,5,7,6] => ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,2,3,8,4,6,7,5] => ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,4,2,3,6,5,8,7,10,9] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [1,4,2,3,6,5,9,10,7,8] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [1,4,2,3,7,8,5,10,6,9] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [1,4,2,3,7,9,5,6,10,8] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [1,4,2,3,8,9,10,5,6,7] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [1,5,6,2,3,8,4,7,10,9] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [1,5,6,2,3,9,10,4,7,8] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [1,5,7,2,3,10,4,8,6,9] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [1,5,7,2,3,4,9,6,10,8] => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [1,5,8,2,3,4,9,10,6,7] => ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [1,6,7,8,2,3,10,4,5,9] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [1,6,7,9,2,3,4,10,5,8] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [1,6,8,9,2,3,4,5,10,7] => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [1,7,8,9,10,2,3,4,5,6] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [1,6,2,3,4,5,8,7,10,9] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [1,6,2,3,4,5,9,10,7,8] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [1,7,8,2,3,4,5,10,6,9] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [1,7,9,2,3,4,5,6,10,8] => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [1,8,9,10,2,3,4,5,6,7] => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [1,8,2,6,3,5,4,7,10,9] => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [1,9,10,2,6,3,5,4,7,8] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [1,10,2,7,3,5,4,8,6,9] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => [1,2,7,3,5,4,9,6,10,8] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => [1,2,8,3,5,4,9,10,6,7] => ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [1,10,2,7,8,3,6,4,5,9] => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => [1,2,7,9,3,6,4,10,5,8] => ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [1,2,8,9,3,6,4,5,10,7] => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [3,7,2,8,9,10,6,5,4,1] => [1,2,8,9,10,3,7,4,5,6] => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [1,8,2,3,4,5,6,7,10,9] => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [1,9,10,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [1,10,2,8,3,4,5,7,6,9] => ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => [1,2,9,3,4,5,7,6,10,8] => ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => [1,2,9,10,3,4,5,8,6,7] => ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [1,10,2,3,8,4,6,7,5,9] => ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [1,2,10,3,9,4,6,7,5,8] => ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => [1,2,3,9,4,6,8,5,10,7] => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [1,2,3,9,10,4,7,8,5,6] => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [5,6,7,9,4,3,2,10,8,1] => [1,2,10,3,4,5,6,7,9,8] => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [1,2,3,10,4,5,6,8,9,7] => ? = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => [1,2,3,4,10,5,7,8,9,6] => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [1,4,2,3,8,10,11,5,6,7,12,9] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => [1,7,9,10,11,2,3,4,5,6,12,8] => ? = 2 - 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000635The number of strictly order preserving maps of a poset into itself. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St000842The breadth of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St000153The number of adjacent cycles of a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation.
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