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Your data matches 321 different statistics following compositions of up to 3 maps.
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Matching statistic: St000981
(load all 10 compositions to match this statistic)
(load all 10 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 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 6 = 5 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 8 = 7 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
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 8 compositions to match this statistic)
(load all 8 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 + 1
[1,0,1,0]
=> 1010 => 4 = 3 + 1
[1,1,0,0]
=> 1100 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 6 = 5 + 1
[1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[1,1,0,0,1,0]
=> 110010 => 3 = 2 + 1
[1,1,0,1,0,0]
=> 110100 => 4 = 3 + 1
[1,1,1,0,0,0]
=> 111000 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 6 = 5 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 2 = 1 + 1
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 6 compositions to match this statistic)
(load all 6 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 + 1
[1,0,1,0]
=> 1010 => 1111 => 4 = 3 + 1
[1,1,0,0]
=> 1100 => 1001 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 111111 => 6 = 5 + 1
[1,0,1,1,0,0]
=> 101100 => 111001 => 3 = 2 + 1
[1,1,0,0,1,0]
=> 110010 => 100111 => 3 = 2 + 1
[1,1,0,1,0,0]
=> 110100 => 100001 => 4 = 3 + 1
[1,1,1,0,0,0]
=> 111000 => 101101 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 6 = 5 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 2 = 1 + 1
Description
The length of the longest constant subword.
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
[1,0,1,0]
=> 1010 => 1111 => 0111 => 3
[1,1,0,0]
=> 1100 => 1001 => 0001 => 1
[1,0,1,0,1,0]
=> 101010 => 111111 => 011111 => 5
[1,0,1,1,0,0]
=> 101100 => 111001 => 011001 => 2
[1,1,0,0,1,0]
=> 110010 => 100111 => 000111 => 3
[1,1,0,1,0,0]
=> 110100 => 100001 => 000001 => 1
[1,1,1,0,0,0]
=> 111000 => 101101 => 001101 => 2
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 01111111 => 7
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 01111001 => 4
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 01100111 => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 01100001 => 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 01101101 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 00011111 => 5
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 00011001 => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 00000111 => 3
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 00000001 => 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 00001101 => 2
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 00110111 => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 00110001 => 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 00111101 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 00100101 => 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
[1,0,1,0]
=> 1010 => 1111 => 0111 => 3
[1,1,0,0]
=> 1100 => 1001 => 0001 => 1
[1,0,1,0,1,0]
=> 101010 => 111111 => 011111 => 5
[1,0,1,1,0,0]
=> 101100 => 111001 => 011001 => 2
[1,1,0,0,1,0]
=> 110010 => 100111 => 000111 => 3
[1,1,0,1,0,0]
=> 110100 => 100001 => 000001 => 1
[1,1,1,0,0,0]
=> 111000 => 101101 => 001101 => 2
[1,0,1,0,1,0,1,0]
=> 10101010 => 11111111 => 01111111 => 7
[1,0,1,0,1,1,0,0]
=> 10101100 => 11111001 => 01111001 => 4
[1,0,1,1,0,0,1,0]
=> 10110010 => 11100111 => 01100111 => 3
[1,0,1,1,0,1,0,0]
=> 10110100 => 11100001 => 01100001 => 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 11101101 => 01101101 => 2
[1,1,0,0,1,0,1,0]
=> 11001010 => 10011111 => 00011111 => 5
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 00011001 => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 10000111 => 00000111 => 3
[1,1,0,1,0,1,0,0]
=> 11010100 => 10000001 => 00000001 => 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10001101 => 00001101 => 2
[1,1,1,0,0,0,1,0]
=> 11100010 => 10110111 => 00110111 => 3
[1,1,1,0,0,1,0,0]
=> 11100100 => 10110001 => 00110001 => 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 10111101 => 00111101 => 4
[1,1,1,1,0,0,0,0]
=> 11110000 => 10100101 => 00100101 => 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: 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 + 1
[1,0,1,0]
=> 1010 => [1,1,1,1] => [4] => 4 = 3 + 1
[1,1,0,0]
=> 1100 => [2,2] => [1,2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => [6] => 6 = 5 + 1
[1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => [3,2,1] => 3 = 2 + 1
[1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => [1,2,3] => 3 = 2 + 1
[1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => [1,4,1] => 4 = 3 + 1
[1,1,1,0,0,0]
=> 111000 => [3,3] => [1,1,2,1,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => [8] => 8 = 7 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => [5,2,1] => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => [3,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => [3,4,1] => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => [3,1,2,1,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => [1,2,5] => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => [1,2,2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => [2,1,1,2,1,1] => [1,4,3] => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => [1,6,1] => 6 = 5 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => [2,1,2,3] => [1,3,2,1,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => [1,1,2,1,3] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => [1,1,2,3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => [1,1,4,1,1] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => [1,1,1,2,1,1,1] => 2 = 1 + 1
Description
The largest part of an integer composition.
Matching statistic: St000444
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 3
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 5
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 7
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000668
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> ? ∊ {1,3}
[1,1,0,0]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> 3
[1,0,1,1,0,0]
=> [1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> [1]
=> ? ∊ {2,5}
[1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {2,5}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,3]
=> 3
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [5]
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [4,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> 4
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> ? ∊ {2,7}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {2,7}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000937
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> ? ∊ {1,3}
[1,1,0,0]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> ? ∊ {3,5}
[1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {3,5}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 5
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 4
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> ? ∊ {3,7}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {3,7}
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000770
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[]]
=> [1]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> [1]
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> [2]
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> [1]
=> ? ∊ {3,5}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> []
=> ? ∊ {3,5}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> [2]
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> [1]
=> ? ∊ {5,7}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> [2,1]
=> 4
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> [3]
=> 3
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> []
=> ? ∊ {5,7}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> [3]
=> 3
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> [2,2]
=> 2
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> [3]
=> 3
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
The following 311 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St000031The number of cycles in the cycle decomposition of a permutation. St000648The number of 2-excedences of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000075The orbit size of a standard tableau under promotion. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000684The global dimension of the LNakayama algebra associated to a Dyck path. 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. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. 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 $eA$. St000455The second largest eigenvalue 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. St001862The number of crossings of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000766The number of inversions of an integer composition. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St001118The acyclic chromatic index of a graph. 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. 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. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. 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. St001403The number of vertical separators in a permutation. 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. St001267The length of the Lyndon factorization of the binary word. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000014The number of parking functions supported by a Dyck path. St000144The pyramid weight of the Dyck path. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000418The number of Dyck paths that are weakly below a Dyck path. St000438The position of the last up step in a Dyck path. St000439The position of the first down step of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We 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 $c_{n−1}$. 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001060The distinguishing index of a graph. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. 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 St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001523The degree of symmetry of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001733The number of weak left to right maxima of a Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001885The number of binary words with the same proper border set. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001959The product of the heights of the peaks of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001330The hat guessing number of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000527The width of the poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000909The number of maximal chains of maximal size in a poset. St000997The even-odd crank of an integer partition. St001618The cardinality of the Frattini sublattice of a lattice. St001964The interval resolution global dimension of a poset. St001645The pebbling number of a connected graph. St000070The number of antichains in a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. 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. St000255The number of reduced Kogan faces with the permutation as type. St000264The girth of a graph, which is not a tree. 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. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. 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. St001346The number of parking functions that give the same permutation. St001388The number of non-attacking neighbors of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001842The major index of a set partition. 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. St000058The order of a permutation. St000089The absolute variation of a composition. 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. 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. St000401The size of the symmetry class of a permutation. 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. St000485The length of the longest cycle of a permutation. St000498The lcs statistic of a set partition. St000502The number of successions of a set partitions. 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. St000638The number of up-down runs of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000670The reversal length of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. 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. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001058The breadth of the ordered tree. St001083The number of boxed occurrences of 132 in a permutation. St001209The pmaj statistic of a parking function. 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. St001402The number of separators in a permutation. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. 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. 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. St001893The flag descent 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. St000060The greater neighbor of the maximum. St000210Minimum over maximum difference of elements in cycles. 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. 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. St001343The dimension of the reduced incidence algebra of a poset. St001375The pancake length of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001516The number of cyclic bonds of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000064The number of one-box pattern of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000542The number of left-to-right-minima of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation.
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