searching the database
Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000245
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 3
Description
The number of ascents of a permutation.
Matching statistic: St000250
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 80%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> {{1}}
=> ? = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 0 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 3 = 1 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 5 = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 5 = 3 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 5 = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 5 = 3 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 5 = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 6 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> {{1,3,4,7},{2},{5},{6},{8}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1,4,6,8},{2,3},{5},{7}}
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> {{1,4,6},{2},{3},{5},{7,8}}
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5,6},{7}}
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> {{1},{2,8},{3,5,7},{4},{6}}
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> {{1,3,6,8},{2},{4,5},{7}}
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,6,8},{4,5},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4},{5,6,8},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> {{1,8},{2,5,7},{3,4},{6}}
=> ? = 4 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,3,4,5,6,8},{2},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,4,5,6,8},{2,3},{7}}
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> {{1,6,8},{2,3,4,5},{7}}
=> ? = 5 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,7,8},{2,3},{4},{5},{6}}
=> ? = 6 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2,7,8},{3,4},{5},{6}}
=> ? = 5 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1,3,5,7,8},{2},{4},{6}}
=> ? = 4 + 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> {{1},{2},{3,7,8},{4,5},{6}}
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4,5},{6},{7,8}}
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> {{1,8},{2,4,6,7},{3},{5}}
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1},{2},{3,4,5},{6},{7,8}}
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,5,6},{3},{4},{8}}
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> {{1,2,7,8},{3,4,5},{6}}
=> ? = 6 + 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,3,4,5,7,8},{2},{6}}
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> {{1,5,7,8},{2,3,4},{6}}
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 6 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> {{1,3,8},{2},{4},{5},{6,7}}
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3},{4},{5,7},{6},{8}}
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> {{1,8},{2,3},{4},{5,7},{6}}
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> {{1},{2},{3,6},{4},{5},{7},{8}}
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,6},{4},{5},{7},{8}}
=> ? = 5 + 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5,7},{6}}
=> ? = 3 + 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> {{1},{2,8},{3,4},{5,7},{6}}
=> ? = 4 + 2
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> {{1,8},{2,6},{3,4},{5},{7}}
=> ? = 5 + 2
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St001492
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St001492: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> 2 = 0 + 2
[1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 4 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 + 2
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St001211
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 2 = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 2 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 2 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 2 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[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]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[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]
=> 5 = 3 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 4 + 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 + 2
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 4 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 + 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5 + 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 5 + 2
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St000105
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => {{1}}
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [8,2,3,4,5,6,1,7] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [8,2,3,4,5,1,7,6] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [8,2,3,4,5,1,6,7] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => [8,2,3,4,1,6,7,5] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [8,2,3,4,7,6,5,1] => ?
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,4,7,8,5,6] => [3,2,1,4,8,7,6,5] => {{1,3},{2},{4},{5,8},{6,7}}
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,4,3,7,5,6,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => ? => ?
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,8,7] => [4,2,1,3,6,5,8,7] => {{1,4},{2},{3},{5,6},{7,8}}
=> ? = 4 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [2,1,5,7,3,8,4,6] => [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,1,5,3,4,6,7,8] => [2,1,5,3,4,6,7,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,1,3,4,6,5,7,8] => [2,1,3,4,6,5,7,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [2,1,3,6,4,7,8,5] => ? => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,1,6,7,3,4,5,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,3,4,5,7,8,6] => [2,1,3,4,5,8,7,6] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,1,3,4,5,7,6,8] => [2,1,3,4,5,7,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,1,3,4,7,5,6,8] => [2,1,3,4,7,5,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6,8] => [2,1,7,3,4,5,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [1,8,3,4,5,6,7,2] => ?
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6,8] => [1,7,3,4,5,2,6,8] => ?
=> ? = 6 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,8,7] => [1,6,3,4,2,5,8,7] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,7,8,6] => [1,5,3,2,4,8,7,6] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,7,4,8,6] => [1,8,3,2,5,4,7,6] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [1,3,2,7,6,5,4,8] => ?
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,2,4,7,8,5] => [1,8,3,2,4,6,7,5] => ?
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,6,7,8,2,4,5] => [1,8,3,7,6,5,4,2] => ?
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,6,2,4,5,7,8] => [1,6,3,2,4,5,7,8] => ?
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,7},{5},{6},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,8,2,4,5,6,7] => [1,8,3,2,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6,8] => [5,1,3,4,2,7,6,8] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [8,3,2,4,5,6,7,1] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,1,5,7,2,6,8] => [7,3,2,4,5,1,6,8] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => [8,5,3,4,2,6,7,1] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [8,7,3,4,5,6,2,1] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,5,8,1,2,6,7] => [8,5,3,4,2,1,6,7] => ?
=> ? = 5 + 1
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000925
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [8,2,3,4,5,6,1,7] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [8,2,3,4,5,1,7,6] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [8,2,3,4,5,1,6,7] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => [8,2,3,4,1,6,7,5] => ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [8,2,3,4,7,6,5,1] => ?
=> ? = 5 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,4,7,8,5,6] => [3,2,1,4,8,7,6,5] => {{1,3},{2},{4},{5,8},{6,7}}
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,4,3,7,5,6,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => ? => ?
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,8,7] => [4,2,1,3,6,5,8,7] => {{1,4},{2},{3},{5,6},{7,8}}
=> ? = 4 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,1,5,6,3,4,8,7] => [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [2,1,5,7,3,8,4,6] => [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,1,5,3,4,6,7,8] => [2,1,5,3,4,6,7,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,1,3,4,6,5,7,8] => [2,1,3,4,6,5,7,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [2,1,3,6,4,7,8,5] => ? => ?
=> ? = 5 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,1,6,7,3,4,5,8] => ? => ?
=> ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,3,4,5,7,8,6] => [2,1,3,4,5,8,7,6] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,1,3,4,5,7,6,8] => [2,1,3,4,5,7,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,1,3,4,7,5,6,8] => [2,1,3,4,7,5,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6,8] => [2,1,7,3,4,5,6,8] => ?
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [1,8,3,4,5,6,7,2] => ?
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6,8] => [1,7,3,4,5,2,6,8] => ?
=> ? = 6 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,8,7] => [1,6,3,4,2,5,8,7] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,7,8,6] => [1,5,3,2,4,8,7,6] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,7,4,8,6] => [1,8,3,2,5,4,7,6] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,2,6,7,4,5,8] => [1,3,2,7,6,5,4,8] => ?
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,3,6,2,4,7,8,5] => [1,8,3,2,4,6,7,5] => ?
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,6,7,8,2,4,5] => [1,8,3,7,6,5,4,2] => ?
=> ? = 4 + 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,6,2,4,5,7,8] => [1,6,3,2,4,5,7,8] => ?
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6,8] => [1,3,2,7,4,5,6,8] => {{1},{2,3},{4,7},{5},{6},{8}}
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => ?
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,8,2,4,5,6,7] => [1,8,3,2,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6,8] => [5,1,3,4,2,7,6,8] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [8,3,2,4,5,6,7,1] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,1,5,7,2,6,8] => [7,3,2,4,5,1,6,8] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => [8,5,3,4,2,6,7,1] => ?
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [8,7,3,4,5,6,2,1] => ?
=> ? = 5 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [2,4,5,1,3] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [2,4,3,5,1] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [2,5,1,3,4] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,4,1,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [3,2,4,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [3,2,5,1,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,2,3,1,6,5,7] => [2,3,4,1,6,5,7] => ? = 4
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [6,2,3,4,1,5,7] => [2,3,4,6,1,5,7] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => [2,3,1,5,6,4,7] => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [6,2,3,1,5,4,7] => [2,3,5,6,1,4,7] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [6,2,3,5,4,1,7] => [2,3,5,4,6,1,7] => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2,3,1,4,6,7] => [2,3,5,1,4,6,7] => ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => [2,3,1,6,4,5,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [7,2,3,1,4,6,5] => [2,3,6,7,1,4,5] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [6,2,3,1,4,5,7] => [2,3,6,1,4,5,7] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => [2,3,7,1,4,5,6] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,4,6,3,5,7] => ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [2,4,5,6,7,1,3] => ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [6,2,1,4,5,3,7] => [2,4,5,6,1,3,7] => ? = 5
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [2,4,5,1,3,6,7] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,5,1,6,3,7] => [6,2,4,3,5,1,7] => [2,4,3,5,6,1,7] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,2,5,4,3,6,1] => [2,4,5,3,6,7,1] => ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => [6,2,5,4,3,1,7] => [2,4,5,3,6,1,7] => ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,5,4,3,1,6] => [2,4,5,3,7,1,6] => ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,4,3,1,6,7] => [2,4,3,5,1,6,7] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [2,4,1,3,6,5,7] => ? = 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [6,2,1,4,3,5,7] => [2,4,6,1,3,5,7] => ? = 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [6,2,4,3,1,5,7] => [2,4,3,6,1,5,7] => ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => [2,1,5,6,3,4,7] => ? = 4
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [2,1,5,4,6,3,7] => ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => [2,5,6,1,3,4,7] => ? = 5
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [6,2,1,5,4,3,7] => [2,5,4,6,1,3,7] => ? = 4
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [6,2,5,3,4,1,7] => [2,5,3,4,6,1,7] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => [1,3,4,5,6,7,2] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => [1,3,4,5,6,2,7] => ? = 5
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => [1,3,4,5,2,6,7] => ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => [1,3,4,6,2,5,7] => ? = 5
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => ? = 5
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [1,3,2,5,6,4,7] => ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,7,4,6] => [1,3,2,7,5,4,6] => [1,3,2,5,7,4,6] => ? = 4
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,7] => [1,6,3,2,5,4,7] => [1,3,5,6,2,4,7] => ? = 5
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,6,2,4,7] => [1,6,3,5,4,2,7] => [1,3,5,4,6,2,7] => ? = 4
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => [1,3,5,2,4,6,7] => ? = 5
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000702
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 70%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,2,4,3] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [5,1,2,4,3] => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [5,1,4,3,2] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => [7,1,3,4,5,6,2] => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,3] => [1,7,2,4,5,6,3] => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => [7,1,2,4,5,6,3] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [7,2,3,1,5,6,4] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,5,2,6,7,4] => [1,7,3,2,5,6,4] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => [7,4,3,2,5,6,1] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,4,5,2,6,7,3] => [1,7,4,3,5,6,2] => ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,1,3,6,7,4] => [7,2,1,3,5,6,4] => ? = 5 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,3,6,7,4] => [1,7,2,3,5,6,4] => ? = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => [7,1,2,3,5,6,4] => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6] => [1,5,2,4,3,7,6] => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [7,2,3,4,1,6,5] => ? = 5 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,6,2,7,5] => [1,7,3,4,2,6,5] => ? = 5 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,5,6,2,7,4] => [1,7,3,5,4,6,2] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,2,5,4,3,6,1] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,4,5,6,2,7,3] => [1,7,5,4,3,6,2] => ? = 4 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => [7,1,5,4,3,6,2] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,2,6,3,7,5] => [1,7,2,4,3,6,5] => ? = 5 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => [1,7,2,5,4,6,3] => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,7,6] => [1,5,3,2,4,7,6] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,4,5,2,3,7,6] => [1,5,4,3,2,7,6] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,6,2,4,7,5] => [1,7,3,2,4,6,5] => ? = 5 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,4,6,2,3,7,5] => [1,7,4,3,2,6,5] => ? = 4 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,6,2,3,7,4] => [1,7,5,3,4,6,2] => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 4 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 4 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => [1,7,2,3,4,6,5] => ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 4 + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [7,1,2,3,4,6,5] => ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => ? = 5 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,6,3,7] => [1,6,2,4,5,3,7] => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5,7] => [4,1,3,2,6,5,7] => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,5,2,6,4,7] => [1,6,3,2,5,4,7] => ? = 5 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,5,2,6,3,7] => [1,6,4,3,5,2,7] => ? = 4 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => [1,6,2,3,5,4,7] => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 4 + 1
[1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,2,4,6,5,7] => [3,1,2,4,6,5,7] => ? = 4 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,1,2,3,6,4,7] => [6,1,2,3,5,4,7] => ? = 5 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,3,4,5,7,2,6] => [1,7,3,4,5,2,6] => ? = 5 + 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,2,5,7,3,6] => [1,7,2,4,5,3,6] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,6,7,2,5] => [1,7,3,4,6,5,2] => ? = 4 + 1
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000213
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,5,7,1] => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,5,1,7] => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ? = 5 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,4,6,1,7] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,5,4,7] => ? = 4 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,6,4,5,7,1] => ? = 5 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,6,4,5,1,7] => ? = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,4,5,7,6] => ? = 4 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ? = 5 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? = 5 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => ? = 5 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => ? = 5 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,4,5,3,7,1] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,4,5,3,1,7] => ? = 4 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,7,4,5,3,6,1] => ? = 4 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,3,6,5,1,7] => ? = 5 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => ? = 4 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => ? = 5 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 4 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => ? = 5 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => ? = 4 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ? = 4 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,3,4,1,6,7] => ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 4 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,6,4,5,3,7] => ? = 4 + 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,1,4,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,3,4,1,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [4,3,1,5,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,3,5,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,5,2,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [4,3,5,1,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,1,4,5,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,5,4,2,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,4,3,5,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,4,3,2] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,4,1,2] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,4,3,1,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,3,4,5,6,7,2] => ? = 5
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => [7,3,4,5,6,2,1] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [7,3,4,5,6,1,2] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [5,2,3,4,1,6,7] => [6,3,4,5,2,7,1] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,2,3,1,6,5,7] => [5,3,4,2,7,6,1] => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [7,2,3,4,1,6,5] => [6,3,4,5,1,7,2] => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [6,2,3,4,1,5,7] => [6,3,4,5,7,2,1] => ? = 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => [6,3,4,5,7,1,2] => ? = 5
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [4,2,3,1,5,6,7] => [5,3,4,2,6,7,1] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => [4,3,2,7,6,5,1] => ? = 4
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [4,3,2,6,5,7,1] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [7,2,3,1,5,6,4] => [5,3,4,1,6,7,2] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [6,2,3,1,5,4,7] => [5,3,4,7,6,2,1] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,2,3,6,5,4,1] => [1,3,4,7,6,5,2] => ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [6,2,3,5,4,1,7] => [7,3,4,6,5,2,1] => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [5,2,3,1,4,6,7] => [5,3,4,6,2,7,1] => ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => [4,3,2,6,7,5,1] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [7,2,3,1,4,6,5] => [5,3,4,6,1,7,2] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [6,2,3,1,4,5,7] => [5,3,4,6,7,2,1] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [4,3,2,5,6,1,7] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => [5,3,4,6,7,1,2] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => [4,3,2,5,6,7,1] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [3,2,7,5,6,4,1] => ? = 4
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [3,2,6,5,4,7,1] => ? = 4
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [3,2,6,5,7,4,1] => ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [3,2,5,4,6,7,1] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [4,3,1,5,6,7,2] => ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,5,6,3,7] => [6,2,1,4,5,3,7] => [4,3,7,5,6,2,1] => ? = 5
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [4,3,6,5,2,7,1] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,5,1,6,3,7] => [6,2,4,3,5,1,7] => [7,3,5,4,6,2,1] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,2,5,4,3,6,1] => [1,3,6,5,4,7,2] => ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => [6,2,5,4,3,1,7] => [7,3,6,5,4,2,1] => ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,5,4,3,1,6] => [7,3,6,5,4,1,2] => ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,4,3,1,6,7] => [6,3,5,4,2,7,1] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,5,2,7,6,1] => ? = 4
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [6,2,1,4,3,5,7] => [4,3,6,5,7,2,1] => ? = 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [6,2,4,3,1,5,7] => [6,3,5,4,7,2,1] => ? = 4
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [4,3,5,2,6,7,1] => ? = 5
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [3,2,4,6,5,1,7] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [3,2,4,6,5,7,1] => ? = 4
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => [3,2,5,7,6,4,1] => ? = 4
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [3,2,7,6,5,4,1] => ? = 3
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [3,2,5,6,4,7,1] => ? = 4
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => [4,3,5,7,6,2,1] => ? = 5
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [6,2,1,5,4,3,7] => [4,3,7,6,5,2,1] => ? = 4
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [6,2,5,3,4,1,7] => [7,3,5,6,4,2,1] => ? = 4
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => [4,3,5,6,2,7,1] => ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [3,2,4,5,1,7,6] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [3,2,4,5,7,6,1] => ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,6,3,4,5,7] => [2,1,6,3,4,5,7] => [3,2,5,6,7,4,1] => ? = 4
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!