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Your data matches 228 different statistics following compositions of up to 3 maps.
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Matching statistic: St000981
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000981: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000981: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 4
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 8
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
Description
The length of the longest zigzag subpath.
This is the length of the longest consecutive subpath that is a zigzag of the form $010...$ or of the form $101...$.
Matching statistic: St000983
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 2
[1,0,1,0]
=> 1010 => 4
[1,1,0,0]
=> 1100 => 2
[1,0,1,0,1,0]
=> 101010 => 6
[1,0,1,1,0,0]
=> 101100 => 3
[1,1,0,0,1,0]
=> 110010 => 3
[1,1,0,1,0,0]
=> 110100 => 4
[1,1,1,0,0,0]
=> 111000 => 2
[1,0,1,0,1,0,1,0]
=> 10101010 => 8
[1,0,1,0,1,1,0,0]
=> 10101100 => 5
[1,0,1,1,0,0,1,0]
=> 10110010 => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 4
[1,0,1,1,1,0,0,0]
=> 10111000 => 3
[1,1,0,0,1,0,1,0]
=> 11001010 => 5
[1,1,0,0,1,1,0,0]
=> 11001100 => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 4
[1,1,0,1,0,1,0,0]
=> 11010100 => 6
[1,1,0,1,1,0,0,0]
=> 11011000 => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 3
[1,1,1,0,1,0,0,0]
=> 11101000 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 2
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000982
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 11 => 2
[1,0,1,0]
=> 1010 => 1111 => 4
[1,1,0,0]
=> 1100 => 1001 => 2
[1,0,1,0,1,0]
=> 101010 => 111111 => 6
[1,0,1,1,0,0]
=> 101100 => 111001 => 3
[1,1,0,0,1,0]
=> 110010 => 100111 => 3
[1,1,0,1,0,0]
=> 110100 => 100001 => 4
[1,1,1,0,0,0]
=> 111000 => 101101 => 2
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 8
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 5
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 4
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 3
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 5
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 4
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 6
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 3
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 2
Description
The length of the longest constant subword.
Matching statistic: St000381
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => [1,1] => [2] => 2
[1,0,1,0]
=> 1010 => [1,1,1,1] => [4] => 4
[1,1,0,0]
=> 1100 => [2,2] => [1,2,1] => 2
[1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => [6] => 6
[1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => [3,2,1] => 3
[1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => [1,2,3] => 3
[1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => [1,4,1] => 4
[1,1,1,0,0,0]
=> 111000 => [3,3] => [1,1,2,1,1] => 2
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => [8] => 8
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => [5,2,1] => 5
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => [3,2,3] => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => [3,4,1] => 4
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => [3,1,2,1,1] => 3
[1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => [1,2,5] => 5
[1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => [1,2,2,2,1] => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => [2,1,1,2,1,1] => [1,4,3] => 4
[1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => [1,6,1] => 6
[1,1,0,1,1,0,0,0]
=> 11011000 => [2,1,2,3] => [1,3,2,1,1] => 3
[1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => [1,1,2,1,3] => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => [1,1,2,3,1] => 3
[1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => [1,1,4,1,1] => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => [1,1,1,2,1,1,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000392
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 11 => 01 => 1 = 2 - 1
[1,0,1,0]
=> 1010 => 1111 => 0111 => 3 = 4 - 1
[1,1,0,0]
=> 1100 => 1001 => 0001 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 111111 => 011111 => 5 = 6 - 1
[1,0,1,1,0,0]
=> 101100 => 111001 => 011001 => 2 = 3 - 1
[1,1,0,0,1,0]
=> 110010 => 100111 => 000111 => 3 = 4 - 1
[1,1,0,1,0,0]
=> 110100 => 100001 => 000001 => 1 = 2 - 1
[1,1,1,0,0,0]
=> 111000 => 101101 => 001101 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 01111111 => 7 = 8 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 01111001 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 01100111 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 01100001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 01101101 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 00011111 => 5 = 6 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 00011001 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 00000111 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 00000001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 00001101 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 00110111 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 00110001 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 00111101 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 00100101 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 11 => 01 => 1 = 2 - 1
[1,0,1,0]
=> 1010 => 1111 => 0111 => 3 = 4 - 1
[1,1,0,0]
=> 1100 => 1001 => 0001 => 1 = 2 - 1
[1,0,1,0,1,0]
=> 101010 => 111111 => 011111 => 5 = 6 - 1
[1,0,1,1,0,0]
=> 101100 => 111001 => 011001 => 2 = 3 - 1
[1,1,0,0,1,0]
=> 110010 => 100111 => 000111 => 3 = 4 - 1
[1,1,0,1,0,0]
=> 110100 => 100001 => 000001 => 1 = 2 - 1
[1,1,1,0,0,0]
=> 111000 => 101101 => 001101 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 01111111 => 7 = 8 - 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 01111001 => 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 01100111 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 01100001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 01101101 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 00011111 => 5 = 6 - 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 00011001 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 00000111 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 00000001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 00001101 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 00110111 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 00110001 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 00111101 => 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 00100101 => 1 = 2 - 1
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000031
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 3
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 4
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 3
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => 3
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => 2
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => ? ∊ {3,3,3,4,4,5,5,6}
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => 2
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => 3
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => 4
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 8
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000648
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 0 = 2 - 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> [3,4,1,2] => 2 = 4 - 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 0 = 2 - 2
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => 4 = 6 - 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => 1 = 3 - 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> {{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => 2 = 4 - 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 0 = 2 - 2
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> [3,4,5,6,7,8,1,2] => 6 = 8 - 2
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> [3,4,5,7,6,1,8,2] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> [3,5,4,7,6,8,1,2] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> [3,5,4,6,7,1,8,2] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> [3,6,4,5,1,7,8,2] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> [2,5,4,6,7,8,1,3] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> [2,5,4,7,6,1,8,3] => 0 = 2 - 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> [2,4,5,7,6,8,1,3] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> {{1,2,4,6},{3,5,7,8}}
=> [2,4,5,6,7,1,8,3] => 4 = 6 - 2
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> {{1,2,4,5},{3,6,7,8}}
=> [2,4,6,5,1,7,8,3] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> [2,3,7,5,6,8,1,4] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> {{1,2,3,6},{4,5,7,8}}
=> [2,3,6,5,7,1,8,4] => ? ∊ {3,3,3,3,3,4,5,5} - 2
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> {{1,2,3,5},{4,6,7,8}}
=> [2,3,5,6,1,7,8,4] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> [2,3,4,1,6,7,8,5] => 0 = 2 - 2
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St001087
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 0 = 2 - 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 0 = 2 - 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 4 - 2
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 1 = 3 - 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 0 = 2 - 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 4 = 6 - 2
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => ? ∊ {2,3,3,3,4,5,5,6} - 2
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => 0 = 2 - 2
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 6 = 8 - 2
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, 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 larger than the first entry of the permutation.
Matching statistic: St000022
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => [1,2] => 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 4
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [1,3,5,2,4,6] => 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [1,3,4,2,5,6] => 3
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [1,2,5,3,4,6] => 3
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [1,2,4,3,5,6] => 4
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 6
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [1,3,5,7,2,4,6,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [1,3,5,6,2,4,7,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [1,3,4,7,2,5,6,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [1,3,4,6,2,5,7,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [1,3,4,5,2,6,7,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => [1,2,5,7,3,4,6,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [1,2,5,6,3,4,7,8] => 4
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [1,2,4,7,3,5,6,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [1,2,4,6,3,5,7,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [1,2,4,5,3,6,7,8] => ? ∊ {2,2,3,3,3,3,3,4,5}
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => [1,2,3,7,4,5,6,8] => 4
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => [1,2,3,6,4,5,7,8] => 5
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => [1,2,3,5,4,6,7,8] => 6
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 8
Description
The number of fixed points of a permutation.
The following 218 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000534The number of 2-rises of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000422The energy of a graph, if it is integral. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001862The number of crossings of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000766The number of inversions of an integer composition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001285The number of primes in the column sums of the two line notation of a permutation. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001778The largest greatest common divisor of an element and its image in a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001403The number of vertical separators in a permutation. St000264The girth of a graph, which is not a tree. St000438The position of the last up step in a Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St001875The number of simple modules with projective dimension at most 1. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000259The diameter of a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001618The cardinality of the Frattini sublattice of a lattice. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000527The width of the poset. St000909The number of maximal chains of maximal size in a poset. St001645The pebbling number of a connected graph. St000070The number of antichains in a poset. St001964The interval resolution global dimension of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000058The order of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000401The size of the symmetry class of a permutation. St000485The length of the longest cycle of a permutation. St000502The number of successions of a set partitions. St000638The number of up-down runs of a permutation. St000670The reversal length of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000836The number of descents of distance 2 of a permutation. St001058The breadth of the ordered tree. St001405The number of bonds in a permutation. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001893The flag descent of a signed permutation. St000060The greater neighbor of the maximum. St000078The number of alternating sign matrices whose left key is the permutation. St000136The dinv of a parking function. St000174The flush statistic of a semistandard tableau. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000255The number of reduced Kogan faces with the permutation as type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000490The intertwining number of a set partition. St000528The height of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000565The major index of a set partition. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000647The number of big descents of a permutation. St000691The number of changes of a binary word. St000742The number of big ascents of a permutation after prepending zero. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000831The number of indices that are either descents or recoils. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001246The maximal difference between two consecutive entries of a permutation. St001343The dimension of the reduced incidence algebra of a poset. St001346The number of parking functions that give the same permutation. St001375The pancake length of a permutation. St001388The number of non-attacking neighbors of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001516The number of cyclic bonds of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001722The number of minimal chains with small intervals between a binary word and the top element. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001842The major index of a set partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001926Sparre Andersen's position of the maximum of a signed permutation. St001948The number of augmented double ascents of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000064The number of one-box pattern of a permutation. St000089The absolute variation of a composition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000406The number of occurrences of the pattern 3241 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000498The lcs statistic of a set partition. St000542The number of left-to-right-minima of a permutation. St000572The dimension exponent of a set partition. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000872The number of very big descents of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001209The pmaj statistic of a parking function. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001298The number of repeated entries in the Lehmer code of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001668The number of points of the poset minus the width of the poset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of 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. St001703The villainy of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001817The number of flag weak exceedances of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000037The sign of a permutation.
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