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Matching statistic: St000731
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => 2 = 3 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 2 = 3 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0 = 1 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => 0 = 1 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000752
Mp00079: Set partitions —shape⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 0 = 1 - 1
{{1,2}}
=> [2]
=> 0 = 1 - 1
{{1},{2}}
=> [1,1]
=> 0 = 1 - 1
{{1,2,3}}
=> [3]
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1]
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1]
=> 0 = 1 - 1
{{1},{2,3}}
=> [2,1]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> 2 = 3 - 1
{{1,2,3},{4}}
=> [3,1]
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,2]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [3,1]
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> [3,2]
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [3,2]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 0 = 1 - 1
{{1},{2,3,4,5}}
=> [4,1]
=> 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [2,2,1,1]
=> 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> 0 = 1 - 1
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St000366
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 2 = 3 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => 2 = 3 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,4,3,2] => 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,4,3] => 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,4,3,2,5,6] => 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0 = 1 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001033
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 0 = 1 - 1
{{1,2}}
=> [2] => [1,1,0,0]
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 0 = 1 - 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001172
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 0 = 1 - 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001323
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> 0 = 1 - 1
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
{{1},{2}}
=> [1,2] => ([],2)
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
{{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0 = 1 - 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 0 = 1 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([],6)
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => ([(5,6)],7)
=> 0 = 1 - 1
Description
The independence gap of a graph.
This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph.
In particular, this statistic is $0$ for well covered graphs
Matching statistic: St000346
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> []
=> 1
{{1,2}}
=> [2]
=> [1,1]
=> [1]
=> 1
{{1},{2}}
=> [1,1]
=> [2]
=> []
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [1]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> []
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> []
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
{{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [5]
=> []
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
{{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2
{{1},{2},{3,4},{5,6}}
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 1
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1
Description
The number of coarsenings of a partition.
A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000883
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,3,2] => 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,2,3] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,2,3] => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,2,3] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,2,3,4] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,3,4,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [1,2,3,4] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,2,4,3] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,2,3,4] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,4,3,2,5] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,3,2,4,5] => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,2,3,5,4] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,2,3,4,5] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => 3
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,3,5,4] => 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,2,4,3,5,6] => 2
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => 1
{{1},{2,3},{4},{5,6}}
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => 1
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => 1
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,2,3,5,4,6] => 2
{{1},{2},{3,4},{5,6}}
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => 1
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => 1
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001066
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4,5}}
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1},{2,3,4},{5}}
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
{{1},{2},{3,4,5},{6}}
=> [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3,4,5}}
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1},{2,3,4},{5}}
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
{{1},{2},{3,4,5},{6}}
=> [1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4},{5},{6}}
=> [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
The following 70 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000214The number of adjacencies of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000877The depth of the binary word interpreted as a path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St000359The number of occurrences of the pattern 23-1. St000931The number of occurrences of the pattern UUU in a Dyck path. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000358The number of occurrences of the pattern 31-2. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000732The number of double deficiencies of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001727The number of invisible inversions of a permutation. St000886The number of permutations with the same antidiagonal sums. St000356The number of occurrences of the pattern 13-2. St000538The number of even inversions of a permutation. St000648The number of 2-excedences of a permutation. St000836The number of descents of distance 2 of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St000365The number of double ascents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000355The number of occurrences of the pattern 21-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001377The major index minus the number of inversions of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000837The number of ascents of distance 2 of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000219The number of occurrences of the pattern 231 in a permutation. St001176The size of a partition minus its first part. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree.
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