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Your data matches 353 different statistics following compositions of up to 3 maps.
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Matching statistic: St001252
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => [1,1]
=> 0
[2]
=> 0 => [2] => [2]
=> 1
[1,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[3]
=> 1 => [1,1] => [1,1]
=> 0
[2,1]
=> 01 => [2,1] => [2,1]
=> 1
[1,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[4]
=> 0 => [2] => [2]
=> 1
[3,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[2,2]
=> 00 => [3] => [3]
=> 0
[2,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[5]
=> 1 => [1,1] => [1,1]
=> 0
[4,1]
=> 01 => [2,1] => [2,1]
=> 1
[3,2]
=> 10 => [1,2] => [2,1]
=> 1
[3,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[2,2,1]
=> 001 => [3,1] => [3,1]
=> 0
[2,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[6]
=> 0 => [2] => [2]
=> 1
[5,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[4,2]
=> 00 => [3] => [3]
=> 0
[4,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 1
[3,3]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[3,2,1]
=> 101 => [1,2,1] => [2,1,1]
=> 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[2,2,2]
=> 000 => [4] => [4]
=> 2
[2,2,1,1]
=> 0011 => [3,1,1] => [3,1,1]
=> 0
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 0
[7]
=> 1 => [1,1] => [1,1]
=> 0
[6,1]
=> 01 => [2,1] => [2,1]
=> 1
[5,2]
=> 10 => [1,2] => [2,1]
=> 1
[5,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[4,3]
=> 01 => [2,1] => [2,1]
=> 1
[4,2,1]
=> 001 => [3,1] => [3,1]
=> 0
[4,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 1
[3,3,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[3,2,2]
=> 100 => [1,3] => [3,1]
=> 0
[3,2,1,1]
=> 1011 => [1,2,1,1] => [2,1,1,1]
=> 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[2,2,2,1]
=> 0001 => [4,1] => [4,1]
=> 2
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => [3,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 0
Description
Half the sum of the even parts of a partition.
Matching statistic: St001465
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [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] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,1,1]
=> [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] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,5,6,7,4] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,4,5,6,7,3] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St001587
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => [1,1] => [1,1]
=> 0
[2]
=> 0 => [2] => [2]
=> 1
[1,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[3]
=> 1 => [1,1] => [1,1]
=> 0
[2,1]
=> 01 => [2,1] => [2,1]
=> 1
[1,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[4]
=> 0 => [2] => [2]
=> 1
[3,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[2,2]
=> 00 => [3] => [3]
=> 0
[2,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[5]
=> 1 => [1,1] => [1,1]
=> 0
[4,1]
=> 01 => [2,1] => [2,1]
=> 1
[3,2]
=> 10 => [1,2] => [2,1]
=> 1
[3,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[2,2,1]
=> 001 => [3,1] => [3,1]
=> 0
[2,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[6]
=> 0 => [2] => [2]
=> 1
[5,1]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[4,2]
=> 00 => [3] => [3]
=> 0
[4,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 1
[3,3]
=> 11 => [1,1,1] => [1,1,1]
=> 0
[3,2,1]
=> 101 => [1,2,1] => [2,1,1]
=> 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
[2,2,2]
=> 000 => [4] => [4]
=> 2
[2,2,1,1]
=> 0011 => [3,1,1] => [3,1,1]
=> 0
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 0
[7]
=> 1 => [1,1] => [1,1]
=> 0
[6,1]
=> 01 => [2,1] => [2,1]
=> 1
[5,2]
=> 10 => [1,2] => [2,1]
=> 1
[5,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[4,3]
=> 01 => [2,1] => [2,1]
=> 1
[4,2,1]
=> 001 => [3,1] => [3,1]
=> 0
[4,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 1
[3,3,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 0
[3,2,2]
=> 100 => [1,3] => [3,1]
=> 0
[3,2,1,1]
=> 1011 => [1,2,1,1] => [2,1,1,1]
=> 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[2,2,2,1]
=> 0001 => [4,1] => [4,1]
=> 2
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => [3,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 0
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St000011
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[7]
=> [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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 0 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St001568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001568: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> ? = 0 + 1
[2]
=> 1 = 0 + 1
[1,1]
=> 2 = 1 + 1
[3]
=> 1 = 0 + 1
[2,1]
=> 1 = 0 + 1
[1,1,1]
=> 2 = 1 + 1
[4]
=> 1 = 0 + 1
[3,1]
=> 1 = 0 + 1
[2,2]
=> 1 = 0 + 1
[2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 2 = 1 + 1
[5]
=> 1 = 0 + 1
[4,1]
=> 1 = 0 + 1
[3,2]
=> 1 = 0 + 1
[3,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 1 = 0 + 1
[2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> 2 = 1 + 1
[6]
=> 1 = 0 + 1
[5,1]
=> 1 = 0 + 1
[4,2]
=> 1 = 0 + 1
[4,1,1]
=> 2 = 1 + 1
[3,3]
=> 1 = 0 + 1
[3,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 2 = 1 + 1
[2,2,2]
=> 1 = 0 + 1
[2,2,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 2 = 1 + 1
[7]
=> 1 = 0 + 1
[6,1]
=> 1 = 0 + 1
[5,2]
=> 1 = 0 + 1
[5,1,1]
=> 2 = 1 + 1
[4,3]
=> 1 = 0 + 1
[4,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 2 = 1 + 1
[3,3,1]
=> 1 = 0 + 1
[3,2,2]
=> 1 = 0 + 1
[3,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1]
=> 1 = 0 + 1
[2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 2 = 1 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001085
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3,5,4,1] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,3,4,5,6] => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,3,4,6,5,1] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,5,3,4,1] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,4,5,2,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2,5] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,1,3,4,5,6,7] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,6,4,5,1] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,5,4,2,1] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,5,2,1,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,1,3,2] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,5,2,1] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [4,2,3,5,6,1] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,2,4,5,1,6] => 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [3,2,4,5,6,7,1] => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001139
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> [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]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St001466
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [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] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,1,1]
=> [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] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,5,6,7,4] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,4,5,6,7,3] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
Description
The number of transpositions swapping cyclically adjacent numbers in a permutation.
Put differently, this is the number of adjacent two-cycles in the chord diagram of a permutation.
Matching statistic: St001732
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [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 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [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 + 1
[1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2 = 1 + 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St001061
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [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] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,1,1]
=> [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] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,2,3,4,5,7,6] => 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,2,3,5,6,7,4] => 0
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,4,5,6,7,3] => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
Description
The number of indices that are both descents and recoils of a permutation.
The following 343 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000214The number of adjacencies of a permutation. St000665The number of rafts of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000360The number of occurrences of the pattern 32-1. St000663The number of right floats of a permutation. St000779The tier of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001728The number of invisible descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000864The number of circled entries of the shifted recording tableau of a permutation. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000658The number of rises of length 2 of a Dyck path. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001423The number of distinct cubes in a binary word. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000150The floored half-sum of the multiplicities of a partition. St000234The number of global ascents of a permutation. St000297The number of leading ones in a binary word. St000352The Elizalde-Pak rank of a permutation. St000386The number of factors DDU in a Dyck path. St000390The number of runs of ones in a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000549The number of odd partial sums of an integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000992The alternating sum of the parts of an integer partition. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001172The number of 1-rises at odd height of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001484The number of singletons of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001657The number of twos in an integer partition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000338The number of pixed points of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000237The number of small exceedances. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000007The number of saliances of the permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000353The number of inner valleys of a permutation. St000502The number of successions of a set partitions. St000628The balance of a binary word. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000940The number of characters of the symmetric group whose value on the partition is zero. St000989The number of final rises of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001114The number of odd descents of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001388The number of non-attacking neighbors of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001597The Frobenius rank of a skew partition. St000117The number of centered tunnels of a Dyck path. St000241The number of cyclical small excedances. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000245The number of ascents of a permutation. St000654The first descent of a permutation. St000834The number of right outer peaks of a permutation. St000990The first ascent of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000740The last entry of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001593This is the number of standard Young tableaux of the given shifted shape. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000461The rix statistic of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000120The number of left tunnels of a Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000547The number of even non-empty partial sums of an integer partition. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001214The aft of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000137The Grundy value of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000219The number of occurrences of the pattern 231 in a permutation. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000455The second largest eigenvalue of a graph if it is integral. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St000768The number of peaks in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St000807The sum of the heights of the valleys of the associated bargraph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000805The number of peaks of the associated bargraph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000782The indicator function of whether a given perfect matching is an L & P matching. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001964The interval resolution global dimension of a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001651The Frankl number of a lattice. St000383The last part of an integer composition. St000681The Grundy value of Chomp on Ferrers diagrams. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000259The diameter of a connected graph. 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. St000649The number of 3-excedences of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000356The number of occurrences of the pattern 13-2. St000054The first entry of the permutation. St000091The descent variation of a composition. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000565The major index of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001469The holeyness of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000522The number of 1-protected nodes of a rooted tree. St000839The largest opener of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000521The number of distinct subtrees of an ordered tree. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau.
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