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Your data matches 550 different statistics following compositions of up to 3 maps.
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Matching statistic: St000760
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => 1 = -1 + 2
[1,2] => 1 = -1 + 2
[2,1] => 2 = 0 + 2
[3] => 1 = -1 + 2
[1,1,2] => 1 = -1 + 2
[1,2,1] => 2 = 0 + 2
[1,3] => 1 = -1 + 2
[2,1,1] => 2 = 0 + 2
[3,1] => 2 = 0 + 2
[4] => 1 = -1 + 2
[1,1,1,2] => 1 = -1 + 2
[1,1,2,1] => 2 = 0 + 2
[1,1,3] => 1 = -1 + 2
[1,2,1,1] => 2 = 0 + 2
[1,3,1] => 2 = 0 + 2
[1,4] => 1 = -1 + 2
[2,1,1,1] => 2 = 0 + 2
[3,1,1] => 2 = 0 + 2
[4,1] => 2 = 0 + 2
[5] => 1 = -1 + 2
[1,1,1,1,2] => 1 = -1 + 2
[1,1,1,2,1] => 2 = 0 + 2
[1,1,1,3] => 1 = -1 + 2
[1,1,2,1,1] => 2 = 0 + 2
[1,1,3,1] => 2 = 0 + 2
[1,1,4] => 1 = -1 + 2
[1,2,1,1,1] => 2 = 0 + 2
[1,3,1,1] => 2 = 0 + 2
[1,4,1] => 2 = 0 + 2
[1,5] => 1 = -1 + 2
[2,1,1,1,1] => 2 = 0 + 2
[3,1,1,1] => 2 = 0 + 2
[4,1,1] => 2 = 0 + 2
[5,1] => 2 = 0 + 2
[6] => 1 = -1 + 2
[1,1,1,1,1,2] => 1 = -1 + 2
[1,1,1,1,2,1] => 2 = 0 + 2
[1,1,1,1,3] => 1 = -1 + 2
[1,1,1,2,1,1] => 2 = 0 + 2
[1,1,1,3,1] => 2 = 0 + 2
[1,1,1,4] => 1 = -1 + 2
[1,1,2,1,1,1] => 2 = 0 + 2
[1,1,3,1,1] => 2 = 0 + 2
[1,1,4,1] => 2 = 0 + 2
[1,1,5] => 1 = -1 + 2
[1,2,1,1,1,1] => 2 = 0 + 2
[1,3,1,1,1] => 2 = 0 + 2
[1,4,1,1] => 2 = 0 + 2
[1,5,1] => 2 = 0 + 2
[1,6] => 1 = -1 + 2
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition.
By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
Matching statistic: St000292
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => 10 => 0 = -1 + 1
[1,2] => 110 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 0 = -1 + 1
[1,1,2] => 1110 => 0 = -1 + 1
[1,2,1] => 1101 => 1 = 0 + 1
[1,3] => 1100 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 0 = -1 + 1
[1,1,1,2] => 11110 => 0 = -1 + 1
[1,1,2,1] => 11101 => 1 = 0 + 1
[1,1,3] => 11100 => 0 = -1 + 1
[1,2,1,1] => 11011 => 1 = 0 + 1
[1,3,1] => 11001 => 1 = 0 + 1
[1,4] => 11000 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 0 = -1 + 1
[1,1,1,1,2] => 111110 => 0 = -1 + 1
[1,1,1,2,1] => 111101 => 1 = 0 + 1
[1,1,1,3] => 111100 => 0 = -1 + 1
[1,1,2,1,1] => 111011 => 1 = 0 + 1
[1,1,3,1] => 111001 => 1 = 0 + 1
[1,1,4] => 111000 => 0 = -1 + 1
[1,2,1,1,1] => 110111 => 1 = 0 + 1
[1,3,1,1] => 110011 => 1 = 0 + 1
[1,4,1] => 110001 => 1 = 0 + 1
[1,5] => 110000 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 0 = -1 + 1
[1,1,1,1,1,2] => 1111110 => 0 = -1 + 1
[1,1,1,1,2,1] => 1111101 => 1 = 0 + 1
[1,1,1,1,3] => 1111100 => 0 = -1 + 1
[1,1,1,2,1,1] => 1111011 => 1 = 0 + 1
[1,1,1,3,1] => 1111001 => 1 = 0 + 1
[1,1,1,4] => 1111000 => 0 = -1 + 1
[1,1,2,1,1,1] => 1110111 => 1 = 0 + 1
[1,1,3,1,1] => 1110011 => 1 = 0 + 1
[1,1,4,1] => 1110001 => 1 = 0 + 1
[1,1,5] => 1110000 => 0 = -1 + 1
[1,2,1,1,1,1] => 1101111 => 1 = 0 + 1
[1,3,1,1,1] => 1100111 => 1 = 0 + 1
[1,4,1,1] => 1100011 => 1 = 0 + 1
[1,5,1] => 1100001 => 1 = 0 + 1
[1,6] => 1100000 => 0 = -1 + 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [1,1,0,0]
=> 0 = -1 + 1
[1,2] => [1,0,1,1,0,0]
=> 0 = -1 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = -1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 0 = -1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = -1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = -1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = -1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0 = -1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0 = -1 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 0 = -1 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000761
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [2] => 0 = -1 + 1
[1,2] => [2,1] => 0 = -1 + 1
[2,1] => [1,2] => 1 = 0 + 1
[3] => [3] => 0 = -1 + 1
[1,1,2] => [2,1,1] => 0 = -1 + 1
[1,2,1] => [1,2,1] => 1 = 0 + 1
[1,3] => [3,1] => 0 = -1 + 1
[2,1,1] => [1,1,2] => 1 = 0 + 1
[3,1] => [1,3] => 1 = 0 + 1
[4] => [4] => 0 = -1 + 1
[1,1,1,2] => [2,1,1,1] => 0 = -1 + 1
[1,1,2,1] => [1,2,1,1] => 1 = 0 + 1
[1,1,3] => [3,1,1] => 0 = -1 + 1
[1,2,1,1] => [1,1,2,1] => 1 = 0 + 1
[1,3,1] => [1,3,1] => 1 = 0 + 1
[1,4] => [4,1] => 0 = -1 + 1
[2,1,1,1] => [1,1,1,2] => 1 = 0 + 1
[3,1,1] => [1,1,3] => 1 = 0 + 1
[4,1] => [1,4] => 1 = 0 + 1
[5] => [5] => 0 = -1 + 1
[1,1,1,1,2] => [2,1,1,1,1] => 0 = -1 + 1
[1,1,1,2,1] => [1,2,1,1,1] => 1 = 0 + 1
[1,1,1,3] => [3,1,1,1] => 0 = -1 + 1
[1,1,2,1,1] => [1,1,2,1,1] => 1 = 0 + 1
[1,1,3,1] => [1,3,1,1] => 1 = 0 + 1
[1,1,4] => [4,1,1] => 0 = -1 + 1
[1,2,1,1,1] => [1,1,1,2,1] => 1 = 0 + 1
[1,3,1,1] => [1,1,3,1] => 1 = 0 + 1
[1,4,1] => [1,4,1] => 1 = 0 + 1
[1,5] => [5,1] => 0 = -1 + 1
[2,1,1,1,1] => [1,1,1,1,2] => 1 = 0 + 1
[3,1,1,1] => [1,1,1,3] => 1 = 0 + 1
[4,1,1] => [1,1,4] => 1 = 0 + 1
[5,1] => [1,5] => 1 = 0 + 1
[6] => [6] => 0 = -1 + 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 0 = -1 + 1
[1,1,1,1,2,1] => [1,2,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,3] => [3,1,1,1,1] => 0 = -1 + 1
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 1 = 0 + 1
[1,1,1,3,1] => [1,3,1,1,1] => 1 = 0 + 1
[1,1,1,4] => [4,1,1,1] => 0 = -1 + 1
[1,1,2,1,1,1] => [1,1,1,2,1,1] => 1 = 0 + 1
[1,1,3,1,1] => [1,1,3,1,1] => 1 = 0 + 1
[1,1,4,1] => [1,4,1,1] => 1 = 0 + 1
[1,1,5] => [5,1,1] => 0 = -1 + 1
[1,2,1,1,1,1] => [1,1,1,1,2,1] => 1 = 0 + 1
[1,3,1,1,1] => [1,1,1,3,1] => 1 = 0 + 1
[1,4,1,1] => [1,1,4,1] => 1 = 0 + 1
[1,5,1] => [1,5,1] => 1 = 0 + 1
[1,6] => [6,1] => 0 = -1 + 1
Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St000340
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [1,1,0,0]
=> 1 = -1 + 2
[1,2] => [1,0,1,1,0,0]
=> 1 = -1 + 2
[2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[3] => [1,1,1,0,0,0]
=> 1 = -1 + 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = -1 + 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 1 = -1 + 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[4] => [1,1,1,1,0,0,0,0]
=> 1 = -1 + 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = -1 + 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = -1 + 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = -1 + 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = -1 + 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = -1 + 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = -1 + 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = -1 + 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = -1 + 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = -1 + 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = -1 + 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = -1 + 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = -1 + 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = -1 + 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 0 + 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 0 + 2
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = -1 + 2
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000390
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => 10 => 1 = -1 + 2
[1,2] => 110 => 1 = -1 + 2
[2,1] => 101 => 2 = 0 + 2
[3] => 100 => 1 = -1 + 2
[1,1,2] => 1110 => 1 = -1 + 2
[1,2,1] => 1101 => 2 = 0 + 2
[1,3] => 1100 => 1 = -1 + 2
[2,1,1] => 1011 => 2 = 0 + 2
[3,1] => 1001 => 2 = 0 + 2
[4] => 1000 => 1 = -1 + 2
[1,1,1,2] => 11110 => 1 = -1 + 2
[1,1,2,1] => 11101 => 2 = 0 + 2
[1,1,3] => 11100 => 1 = -1 + 2
[1,2,1,1] => 11011 => 2 = 0 + 2
[1,3,1] => 11001 => 2 = 0 + 2
[1,4] => 11000 => 1 = -1 + 2
[2,1,1,1] => 10111 => 2 = 0 + 2
[3,1,1] => 10011 => 2 = 0 + 2
[4,1] => 10001 => 2 = 0 + 2
[5] => 10000 => 1 = -1 + 2
[1,1,1,1,2] => 111110 => 1 = -1 + 2
[1,1,1,2,1] => 111101 => 2 = 0 + 2
[1,1,1,3] => 111100 => 1 = -1 + 2
[1,1,2,1,1] => 111011 => 2 = 0 + 2
[1,1,3,1] => 111001 => 2 = 0 + 2
[1,1,4] => 111000 => 1 = -1 + 2
[1,2,1,1,1] => 110111 => 2 = 0 + 2
[1,3,1,1] => 110011 => 2 = 0 + 2
[1,4,1] => 110001 => 2 = 0 + 2
[1,5] => 110000 => 1 = -1 + 2
[2,1,1,1,1] => 101111 => 2 = 0 + 2
[3,1,1,1] => 100111 => 2 = 0 + 2
[4,1,1] => 100011 => 2 = 0 + 2
[5,1] => 100001 => 2 = 0 + 2
[6] => 100000 => 1 = -1 + 2
[1,1,1,1,1,2] => 1111110 => 1 = -1 + 2
[1,1,1,1,2,1] => 1111101 => 2 = 0 + 2
[1,1,1,1,3] => 1111100 => 1 = -1 + 2
[1,1,1,2,1,1] => 1111011 => 2 = 0 + 2
[1,1,1,3,1] => 1111001 => 2 = 0 + 2
[1,1,1,4] => 1111000 => 1 = -1 + 2
[1,1,2,1,1,1] => 1110111 => 2 = 0 + 2
[1,1,3,1,1] => 1110011 => 2 = 0 + 2
[1,1,4,1] => 1110001 => 2 = 0 + 2
[1,1,5] => 1110000 => 1 = -1 + 2
[1,2,1,1,1,1] => 1101111 => 2 = 0 + 2
[1,3,1,1,1] => 1100111 => 2 = 0 + 2
[1,4,1,1] => 1100011 => 2 = 0 + 2
[1,5,1] => 1100001 => 2 = 0 + 2
[1,6] => 1100000 => 1 = -1 + 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000691
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => 10 => 1 = -1 + 2
[1,2] => 110 => 1 = -1 + 2
[2,1] => 101 => 2 = 0 + 2
[3] => 100 => 1 = -1 + 2
[1,1,2] => 1110 => 1 = -1 + 2
[1,2,1] => 1101 => 2 = 0 + 2
[1,3] => 1100 => 1 = -1 + 2
[2,1,1] => 1011 => 2 = 0 + 2
[3,1] => 1001 => 2 = 0 + 2
[4] => 1000 => 1 = -1 + 2
[1,1,1,2] => 11110 => 1 = -1 + 2
[1,1,2,1] => 11101 => 2 = 0 + 2
[1,1,3] => 11100 => 1 = -1 + 2
[1,2,1,1] => 11011 => 2 = 0 + 2
[1,3,1] => 11001 => 2 = 0 + 2
[1,4] => 11000 => 1 = -1 + 2
[2,1,1,1] => 10111 => 2 = 0 + 2
[3,1,1] => 10011 => 2 = 0 + 2
[4,1] => 10001 => 2 = 0 + 2
[5] => 10000 => 1 = -1 + 2
[1,1,1,1,2] => 111110 => 1 = -1 + 2
[1,1,1,2,1] => 111101 => 2 = 0 + 2
[1,1,1,3] => 111100 => 1 = -1 + 2
[1,1,2,1,1] => 111011 => 2 = 0 + 2
[1,1,3,1] => 111001 => 2 = 0 + 2
[1,1,4] => 111000 => 1 = -1 + 2
[1,2,1,1,1] => 110111 => 2 = 0 + 2
[1,3,1,1] => 110011 => 2 = 0 + 2
[1,4,1] => 110001 => 2 = 0 + 2
[1,5] => 110000 => 1 = -1 + 2
[2,1,1,1,1] => 101111 => 2 = 0 + 2
[3,1,1,1] => 100111 => 2 = 0 + 2
[4,1,1] => 100011 => 2 = 0 + 2
[5,1] => 100001 => 2 = 0 + 2
[6] => 100000 => 1 = -1 + 2
[1,1,1,1,1,2] => 1111110 => 1 = -1 + 2
[1,1,1,1,2,1] => 1111101 => 2 = 0 + 2
[1,1,1,1,3] => 1111100 => 1 = -1 + 2
[1,1,1,2,1,1] => 1111011 => 2 = 0 + 2
[1,1,1,3,1] => 1111001 => 2 = 0 + 2
[1,1,1,4] => 1111000 => 1 = -1 + 2
[1,1,2,1,1,1] => 1110111 => 2 = 0 + 2
[1,1,3,1,1] => 1110011 => 2 = 0 + 2
[1,1,4,1] => 1110001 => 2 = 0 + 2
[1,1,5] => 1110000 => 1 = -1 + 2
[1,2,1,1,1,1] => 1101111 => 2 = 0 + 2
[1,3,1,1,1] => 1100111 => 2 = 0 + 2
[1,4,1,1] => 1100011 => 2 = 0 + 2
[1,5,1] => 1100001 => 2 = 0 + 2
[1,6] => 1100000 => 1 = -1 + 2
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000758
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [2] => 1 = -1 + 2
[1,2] => [2,1] => 1 = -1 + 2
[2,1] => [1,2] => 2 = 0 + 2
[3] => [3] => 1 = -1 + 2
[1,1,2] => [2,1,1] => 1 = -1 + 2
[1,2,1] => [1,2,1] => 2 = 0 + 2
[1,3] => [3,1] => 1 = -1 + 2
[2,1,1] => [1,1,2] => 2 = 0 + 2
[3,1] => [1,3] => 2 = 0 + 2
[4] => [4] => 1 = -1 + 2
[1,1,1,2] => [2,1,1,1] => 1 = -1 + 2
[1,1,2,1] => [1,2,1,1] => 2 = 0 + 2
[1,1,3] => [3,1,1] => 1 = -1 + 2
[1,2,1,1] => [1,1,2,1] => 2 = 0 + 2
[1,3,1] => [1,3,1] => 2 = 0 + 2
[1,4] => [4,1] => 1 = -1 + 2
[2,1,1,1] => [1,1,1,2] => 2 = 0 + 2
[3,1,1] => [1,1,3] => 2 = 0 + 2
[4,1] => [1,4] => 2 = 0 + 2
[5] => [5] => 1 = -1 + 2
[1,1,1,1,2] => [2,1,1,1,1] => 1 = -1 + 2
[1,1,1,2,1] => [1,2,1,1,1] => 2 = 0 + 2
[1,1,1,3] => [3,1,1,1] => 1 = -1 + 2
[1,1,2,1,1] => [1,1,2,1,1] => 2 = 0 + 2
[1,1,3,1] => [1,3,1,1] => 2 = 0 + 2
[1,1,4] => [4,1,1] => 1 = -1 + 2
[1,2,1,1,1] => [1,1,1,2,1] => 2 = 0 + 2
[1,3,1,1] => [1,1,3,1] => 2 = 0 + 2
[1,4,1] => [1,4,1] => 2 = 0 + 2
[1,5] => [5,1] => 1 = -1 + 2
[2,1,1,1,1] => [1,1,1,1,2] => 2 = 0 + 2
[3,1,1,1] => [1,1,1,3] => 2 = 0 + 2
[4,1,1] => [1,1,4] => 2 = 0 + 2
[5,1] => [1,5] => 2 = 0 + 2
[6] => [6] => 1 = -1 + 2
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 1 = -1 + 2
[1,1,1,1,2,1] => [1,2,1,1,1,1] => 2 = 0 + 2
[1,1,1,1,3] => [3,1,1,1,1] => 1 = -1 + 2
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 2 = 0 + 2
[1,1,1,3,1] => [1,3,1,1,1] => 2 = 0 + 2
[1,1,1,4] => [4,1,1,1] => 1 = -1 + 2
[1,1,2,1,1,1] => [1,1,1,2,1,1] => 2 = 0 + 2
[1,1,3,1,1] => [1,1,3,1,1] => 2 = 0 + 2
[1,1,4,1] => [1,4,1,1] => 2 = 0 + 2
[1,1,5] => [5,1,1] => 1 = -1 + 2
[1,2,1,1,1,1] => [1,1,1,1,2,1] => 2 = 0 + 2
[1,3,1,1,1] => [1,1,1,3,1] => 2 = 0 + 2
[1,4,1,1] => [1,1,4,1] => 2 = 0 + 2
[1,5,1] => [1,5,1] => 2 = 0 + 2
[1,6] => [6,1] => 1 = -1 + 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000764
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000764: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [2] => 1 = -1 + 2
[1,2] => [2,1] => 1 = -1 + 2
[2,1] => [1,2] => 2 = 0 + 2
[3] => [3] => 1 = -1 + 2
[1,1,2] => [2,1,1] => 1 = -1 + 2
[1,2,1] => [1,2,1] => 2 = 0 + 2
[1,3] => [3,1] => 1 = -1 + 2
[2,1,1] => [1,1,2] => 2 = 0 + 2
[3,1] => [1,3] => 2 = 0 + 2
[4] => [4] => 1 = -1 + 2
[1,1,1,2] => [2,1,1,1] => 1 = -1 + 2
[1,1,2,1] => [1,2,1,1] => 2 = 0 + 2
[1,1,3] => [3,1,1] => 1 = -1 + 2
[1,2,1,1] => [1,1,2,1] => 2 = 0 + 2
[1,3,1] => [1,3,1] => 2 = 0 + 2
[1,4] => [4,1] => 1 = -1 + 2
[2,1,1,1] => [1,1,1,2] => 2 = 0 + 2
[3,1,1] => [1,1,3] => 2 = 0 + 2
[4,1] => [1,4] => 2 = 0 + 2
[5] => [5] => 1 = -1 + 2
[1,1,1,1,2] => [2,1,1,1,1] => 1 = -1 + 2
[1,1,1,2,1] => [1,2,1,1,1] => 2 = 0 + 2
[1,1,1,3] => [3,1,1,1] => 1 = -1 + 2
[1,1,2,1,1] => [1,1,2,1,1] => 2 = 0 + 2
[1,1,3,1] => [1,3,1,1] => 2 = 0 + 2
[1,1,4] => [4,1,1] => 1 = -1 + 2
[1,2,1,1,1] => [1,1,1,2,1] => 2 = 0 + 2
[1,3,1,1] => [1,1,3,1] => 2 = 0 + 2
[1,4,1] => [1,4,1] => 2 = 0 + 2
[1,5] => [5,1] => 1 = -1 + 2
[2,1,1,1,1] => [1,1,1,1,2] => 2 = 0 + 2
[3,1,1,1] => [1,1,1,3] => 2 = 0 + 2
[4,1,1] => [1,1,4] => 2 = 0 + 2
[5,1] => [1,5] => 2 = 0 + 2
[6] => [6] => 1 = -1 + 2
[1,1,1,1,1,2] => [2,1,1,1,1,1] => 1 = -1 + 2
[1,1,1,1,2,1] => [1,2,1,1,1,1] => 2 = 0 + 2
[1,1,1,1,3] => [3,1,1,1,1] => 1 = -1 + 2
[1,1,1,2,1,1] => [1,1,2,1,1,1] => 2 = 0 + 2
[1,1,1,3,1] => [1,3,1,1,1] => 2 = 0 + 2
[1,1,1,4] => [4,1,1,1] => 1 = -1 + 2
[1,1,2,1,1,1] => [1,1,1,2,1,1] => 2 = 0 + 2
[1,1,3,1,1] => [1,1,3,1,1] => 2 = 0 + 2
[1,1,4,1] => [1,4,1,1] => 2 = 0 + 2
[1,1,5] => [5,1,1] => 1 = -1 + 2
[1,2,1,1,1,1] => [1,1,1,1,2,1] => 2 = 0 + 2
[1,3,1,1,1] => [1,1,1,3,1] => 2 = 0 + 2
[1,4,1,1] => [1,1,4,1] => 2 = 0 + 2
[1,5,1] => [1,5,1] => 2 = 0 + 2
[1,6] => [6,1] => 1 = -1 + 2
Description
The number of strong records in an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Matching statistic: St000159
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2] => [[2],[]]
=> []
=> 0 = -1 + 1
[1,2] => [[2,1],[]]
=> []
=> 0 = -1 + 1
[2,1] => [[2,2],[1]]
=> [1]
=> 1 = 0 + 1
[3] => [[3],[]]
=> []
=> 0 = -1 + 1
[1,1,2] => [[2,1,1],[]]
=> []
=> 0 = -1 + 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,3] => [[3,1],[]]
=> []
=> 0 = -1 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> 1 = 0 + 1
[4] => [[4],[]]
=> []
=> 0 = -1 + 1
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,3] => [[3,1,1],[]]
=> []
=> 0 = -1 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 0 + 1
[1,4] => [[4,1],[]]
=> []
=> 0 = -1 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> 1 = 0 + 1
[5] => [[5],[]]
=> []
=> 0 = -1 + 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1 = 0 + 1
[1,1,4] => [[4,1,1],[]]
=> []
=> 0 = -1 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 = 0 + 1
[1,5] => [[5,1],[]]
=> []
=> 0 = -1 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> 1 = 0 + 1
[6] => [[6],[]]
=> []
=> 0 = -1 + 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 1 = 0 + 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> 0 = -1 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> 1 = 0 + 1
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> 1 = 0 + 1
[1,1,5] => [[5,1,1],[]]
=> []
=> 0 = -1 + 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> 1 = 0 + 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> 1 = 0 + 1
[1,5,1] => [[5,5,1],[4]]
=> [4]
=> 1 = 0 + 1
[1,6] => [[6,1],[]]
=> []
=> 0 = -1 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
The following 540 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000291The number of descents of a binary word. St000768The number of peaks in an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000068The number of minimal elements in a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000767The number of runs in an integer composition. St001486The number of corners of the ribbon associated with an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000052The number of valleys of a Dyck path not on the x-axis. St000065The number of entries equal to -1 in an alternating sign matrix. St000257The number of distinct parts of a partition that occur at least twice. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000769The major index of a composition regarded as a word. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001673The degree of asymmetry of an integer composition. St001691The number of kings in a graph. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000011The number of touch points (or returns) of a Dyck path. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000157The number of descents of a standard tableau. St000273The domination number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000388The number of orbits of vertices of a graph under automorphisms. St000442The maximal area to the right of an up step of a Dyck path. St000527The width of the poset. St000544The cop number of a graph. St000553The number of blocks of a graph. St000568The hook number of a binary tree. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000733The row containing the largest entry of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000783The side length of the largest staircase partition fitting into a partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000916The packing number of a graph. St000920The logarithmic height of a Dyck path. St001432The order dimension of the partition. St001732The number of peaks visible from the left. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001829The common independence number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000010The length of the partition. St000013The height of a Dyck path. St000097The order of the largest clique of the graph. St000288The number of ones in a binary word. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000918The 2-limited packing number of a graph. St001581The achromatic number of a graph. St000806The semiperimeter of the associated bargraph. St001271The competition number of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000535The rank-width of a graph. St000948The chromatic discriminant of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000053The number of valleys of the Dyck path. St000201The number of leaf nodes in a binary tree. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000204The number of internal nodes of a binary tree. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000497The lcb statistic of a set partition. St000552The number of cut vertices of a graph. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000932The number of occurrences of the pattern UDU in a Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001689The number of celebrities in a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000069The number of maximal elements of a poset. St000272The treewidth of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000362The size of a minimal vertex cover of a graph. St000522The number of 1-protected nodes of a rooted tree. St000526The number of posets with combinatorially isomorphic order polytopes. St000528The height of a poset. St000536The pathwidth of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000717The number of ordinal summands of a poset. St000846The maximal number of elements covering an element of a poset. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001358The largest degree of a regular subgraph of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001463The number of distinct columns in the nullspace of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001717The largest size of an interval in a poset. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. 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 $c_{n-1}$. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St000315The number of isolated vertices of a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001367The smallest number which does not occur as degree of a vertex in a graph. St000287The number of connected components of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St000096The number of spanning trees of a graph. St000080The rank of the poset. St000149The number of cells of the partition whose leg is zero and arm is odd. St000617The number of global maxima of a Dyck path. St000647The number of big descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000352The Elizalde-Pak rank of a permutation. St000306The bounce count of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000007The number of saliances of the permutation. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000353The number of inner valleys of a permutation. St000796The stat' of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000360The number of occurrences of the pattern 32-1. St000711The number of big exceedences of a permutation. St001668The number of points of the poset minus the width of the poset. St001728The number of invisible descents of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000470The number of runs in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000646The number of big ascents of a permutation. St000779The tier of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000619The number of cyclic descents of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001735The number of permutations with the same set of runs. St000256The number of parts from which one can substract 2 and still get an integer partition. St001933The largest multiplicity of a part in an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000919The number of maximal left branches of a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000150The floored half-sum of the multiplicities of a partition. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000054The first entry of the permutation. St000260The radius of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St001571The Cartan determinant of the integer partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000145The Dyson rank of a partition. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001592The maximal number of simple paths between any two different vertices of a graph. St000358The number of occurrences of the pattern 31-2. St000710The number of big deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000164The number of short pairs. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000259The diameter of a connected graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000092The number of outer peaks of a permutation. St000331The number of upper interactions of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000638The number of up-down runs of a permutation. St000117The number of centered tunnels of a Dyck path. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000628The balance of a binary word. St000629The defect of a binary word. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000699The toughness times the least common multiple of 1,. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001910The height of the middle non-run of a Dyck path. St000023The number of inner peaks of a permutation. St000091The descent variation of a composition. St000133The "bounce" of a permutation. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000338The number of pixed points of a permutation. St000523The number of 2-protected nodes of a rooted tree. 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. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001091The number of parts in an integer partition whose next smaller part has the same size. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001838The number of nonempty primitive factors of a binary word. St001884The number of borders of a binary word. St000021The number of descents of a permutation. St000099The number of valleys of a permutation, including the boundary. St000238The number of indices that are not small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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 $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001375The pancake length of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001760The number of prefix or suffix reversals needed to sort a permutation. St001812The biclique partition number of a graph. St000264The girth of a graph, which is not a tree. St000325The width of the tree associated to a permutation. St000397The Strahler number of a rooted tree. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. 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. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St000834The number of right outer peaks of a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000369The dinv deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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 $eA$. St001597The Frobenius rank of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000706The product of the factorials of the multiplicities of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000709The number of occurrences of 14-2-3 or 14-3-2. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001394The genus of a permutation. St000703The number of deficiencies of a permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001960The number of descents of a permutation minus one if its first entry is not one. St000478Another weight of a partition according to Alladi. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000934The 2-degree of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St000100The number of linear extensions of a poset. St000252The number of nodes of degree 3 of a binary tree. St000461The rix statistic of a permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001549The number of restricted non-inversions between exceedances. 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. St001731The factorization defect of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000216The absolute length of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of 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. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001060The distinguishing index of a graph. St001330The hat guessing number of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000993The multiplicity of the largest part of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001413Half the length of the longest even length palindromic prefix of a binary word. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001569The maximal modular displacement of a permutation. St001488The number of corners of a skew partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001399The distinguishing number of a poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1.
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