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Your data matches 46 different statistics following compositions of up to 3 maps.
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Matching statistic: St001011
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000291
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ 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] => 10 => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1
Description
The number of descents of a binary word.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,1,3] => [2,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,5,1,2,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => [2,1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,2,4,1,3] => [3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,1,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => [3,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,5,1,3] => [2,1,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001333
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let F be a set of graphs. A set of vertices S is F-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of S does not contain any graph in F.
This statistic returns the cardinality of the smallest isolating set when F contains only the graph with one edge.
Matching statistic: St000390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => => => ? = 0
[1,0,1,0]
=> [2,1] => 0 => 1 => 1
[1,1,0,0]
=> [1,2] => 1 => 0 => 0
[1,0,1,0,1,0]
=> [2,1,3] => 01 => 10 => 1
[1,0,1,1,0,0]
=> [2,3,1] => 00 => 01 => 1
[1,1,0,0,1,0]
=> [3,1,2] => 00 => 01 => 1
[1,1,0,1,0,0]
=> [1,3,2] => 10 => 11 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 010 => 101 => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 000 => 001 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 011 => 100 => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 001 => 010 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 001 => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 000 => 001 => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 001 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 001 => 010 => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 101 => 110 => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 100 => 011 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 100 => 011 => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 110 => 111 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0101 => 1010 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0001 => 0010 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 0100 => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0000 => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0000 => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 0100 => 1001 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0000 => 0001 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 0110 => 1011 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0010 => 0101 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0000 => 0001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 0111 => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0011 => 0100 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0001 => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 0001 => 0010 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0001 => 0010 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 0000 => 0001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0000 => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 0000 => 0001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0000 => 0001 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 0010 => 0101 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 1010 => 1101 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1000 => 0011 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0011 => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1011 => 1100 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 1001 => 0110 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1000 => 0011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 0000 => 0001 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000659
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000374
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [5,3,2,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,4,3,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [4,2,3,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [5,2,4,3,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [5,2,4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,4,3,6,5] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,1,4,3,6,5] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [7,2,4,3,1,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,2,1,4,6,5,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [7,2,4,3,6,5,1] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [7,2,1,4,3,5,6] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [7,2,4,3,1,5,6] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [5,2,1,4,3,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [7,2,1,4,5,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [7,2,4,3,5,1,6] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [7,2,4,3,5,6,1] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => [2,1,6,4,5,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => [6,3,2,4,5,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [2,1,7,3,5,6,4] => [2,1,7,4,6,5,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [7,2,1,3,5,6,4] => [7,3,2,4,6,5,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [7,2,1,5,4,6,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [7,2,1,6,5,4,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,7,4,6] => [2,1,7,3,5,4,6] => [2,1,7,4,6,5,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6] => [7,2,1,3,5,4,6] => [7,3,2,4,6,5,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [7,2,1,5,4,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,5,3,4,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [5,2,1,3,4,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6] => [2,1,3,7,5,4,6] => [2,1,3,7,6,5,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6] => [7,2,3,1,5,4,6] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => [7,2,3,5,4,1,6] => [7,6,3,5,4,2,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation π=[π1,…,πn], this statistic counts the number of position j such that πj<j and there do not exist indices i,k with i<j<k and πi>πj>πk.
See also [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,3,2,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [4,2,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [5,3,2,4,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [5,4,3,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,4,3,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [4,2,3,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [5,2,4,3,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [5,2,4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,4,3,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,4,3,6,5] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,1,4,3,6,5] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [7,2,4,3,1,6,5] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,2,1,4,6,5,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [7,2,4,3,6,5,1] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,6,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,7,5,4,6,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [7,2,1,4,3,5,6] => [7,3,2,5,4,6,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [7,2,4,3,1,5,6] => [7,5,4,3,2,6,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [5,2,1,4,3,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [7,2,1,4,5,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [7,2,4,3,5,1,6] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [7,2,4,3,5,6,1] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => [2,1,6,4,5,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => [6,3,2,4,5,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [2,1,7,3,5,6,4] => [2,1,7,4,6,5,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [7,2,1,3,5,6,4] => [7,3,2,4,6,5,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [7,2,1,5,4,6,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [7,2,1,6,5,4,3] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,7,4,6] => [2,1,7,3,5,4,6] => [2,1,7,4,6,5,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6] => [7,2,1,3,5,4,6] => [7,3,2,4,6,5,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [7,2,1,5,4,3,6] => [7,3,2,6,5,4,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,5,3,4,7,6] => [2,1,5,4,3,7,6] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [5,2,1,3,4,7,6] => [5,3,2,4,1,7,6] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6] => [2,1,3,7,5,4,6] => [2,1,3,7,6,5,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6] => [7,2,3,1,5,4,6] => [7,4,3,2,6,5,1] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => [7,2,3,5,4,1,6] => [7,6,3,5,4,2,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,5,4,3,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,5,1,3,4,6,7] => [5,2,1,3,4,6,7] => [5,3,2,4,1,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,3,5,6,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001665
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => {{1}}
=> [1] => 0
[1,0,1,0]
=> [2,1] => {{1,2}}
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => {{1,2},{3,4},{5,6},{7}}
=> [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => {{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => {{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => {{1,2},{3,4,6},{5,7}}
=> [2,1,4,6,7,3,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => {{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => {{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => {{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => {{1,2},{3,4},{5},{6,7}}
=> [2,1,4,3,5,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => {{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => {{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => {{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => {{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => {{1,2},{3,4},{5},{6},{7}}
=> [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => {{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => {{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => {{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => {{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => {{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => {{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => {{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => {{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => ? = 2
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation π is a position i<πi such that there is no j<i with i≤πj<πi.
Matching statistic: St001859
(load all 177 compositions to match this statistic)
(load all 177 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001859: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[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]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => ? = 1
Description
The number of factors of the Stanley symmetric function associated with a permutation.
For example, the Stanley symmetric function of π=321645 equals
20m1,1,1,1,1+11m2,1,1,1+6m2,2,1+4m3,1,1+2m3,2+m4,1=(m1,1+m2)(2m1,1,1+m2,1).
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001737The number of descents of type 2 in a permutation. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. 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]/(xn). St001545The second Elser number of a connected graph. St001330The hat guessing number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001344The neighbouring number of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St001413Half the length of the longest even length palindromic prefix of a binary word. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St001722The number of minimal chains with small intervals between a binary word and the top element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000028The number of stack-sorts needed to sort a permutation. St000862The number of parts of the shifted shape of a permutation.
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