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Your data matches 151 different statistics following compositions of up to 3 maps.
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Matching statistic: St000660
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001022
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000052
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000768
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1,1] => 0
[1,1] => 11 => [1,1,1] => 0
[2] => 10 => [1,2] => 0
[1,1,1] => 111 => [1,1,1,1] => 0
[1,2] => 110 => [1,1,2] => 0
[2,1] => 101 => [1,2,1] => 1
[3] => 100 => [1,3] => 0
[1,1,1,1] => 1111 => [1,1,1,1,1] => 0
[1,1,2] => 1110 => [1,1,1,2] => 0
[1,2,1] => 1101 => [1,1,2,1] => 1
[1,3] => 1100 => [1,1,3] => 0
[2,1,1] => 1011 => [1,2,1,1] => 1
[2,2] => 1010 => [1,2,2] => 0
[3,1] => 1001 => [1,3,1] => 1
[4] => 1000 => [1,4] => 0
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => 0
[1,1,1,2] => 11110 => [1,1,1,1,2] => 0
[1,1,2,1] => 11101 => [1,1,1,2,1] => 1
[1,1,3] => 11100 => [1,1,1,3] => 0
[1,2,1,1] => 11011 => [1,1,2,1,1] => 1
[1,2,2] => 11010 => [1,1,2,2] => 0
[1,3,1] => 11001 => [1,1,3,1] => 1
[1,4] => 11000 => [1,1,4] => 0
[2,1,1,1] => 10111 => [1,2,1,1,1] => 1
[2,1,2] => 10110 => [1,2,1,2] => 1
[2,2,1] => 10101 => [1,2,2,1] => 0
[2,3] => 10100 => [1,2,3] => 0
[3,1,1] => 10011 => [1,3,1,1] => 1
[3,2] => 10010 => [1,3,2] => 1
[4,1] => 10001 => [1,4,1] => 1
[5] => 10000 => [1,5] => 0
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => 0
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => 0
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => 1
[1,1,1,3] => 111100 => [1,1,1,1,3] => 0
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => 1
[1,1,2,2] => 111010 => [1,1,1,2,2] => 0
[1,1,3,1] => 111001 => [1,1,1,3,1] => 1
[1,1,4] => 111000 => [1,1,1,4] => 0
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => 1
[1,2,1,2] => 110110 => [1,1,2,1,2] => 1
[1,2,2,1] => 110101 => [1,1,2,2,1] => 0
[1,2,3] => 110100 => [1,1,2,3] => 0
[1,3,1,1] => 110011 => [1,1,3,1,1] => 1
[1,3,2] => 110010 => [1,1,3,2] => 1
[1,4,1] => 110001 => [1,1,4,1] => 1
[1,5] => 110000 => [1,1,5] => 0
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => 1
[2,1,1,2] => 101110 => [1,2,1,1,2] => 1
[2,1,2,1] => 101101 => [1,2,1,2,1] => 2
Description
The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St001037
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000257
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 0
[2] => [[2],[]]
=> [2]
=> []
=> 0
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 0
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 0
[3] => [[3],[]]
=> [3]
=> []
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 0
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 1
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 0
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 0
[4] => [[4],[]]
=> [4]
=> []
=> 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 0
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 0
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 1
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 0
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 0
[5] => [[5],[]]
=> [5]
=> []
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 1
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 0
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000386
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000552
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,1] => [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[2] => [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0
[3] => [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 0
Description
The number of cut vertices of a graph.
A cut vertex is one whose deletion increases the number of connected components.
Matching statistic: St001092
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 0
[2] => [[2],[]]
=> [2]
=> []
=> 0
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 0
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 0
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 1
[3] => [[3],[]]
=> [3]
=> []
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 1
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 1
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 0
[4] => [[4],[]]
=> [4]
=> []
=> 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 0
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 0
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 0
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 1
[5] => [[5],[]]
=> [5]
=> []
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 0
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 1
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001167
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001689The number of celebrities in a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001732The number of peaks visible from the left. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001469The holeyness of a permutation. St001737The number of descents of type 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000647The number of big descents of a permutation. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000035The number of left outer peaks of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000836The number of descents of distance 2 of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000360The number of occurrences of the pattern 32-1. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St001727The number of invisible inversions of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001597The Frobenius rank of a skew partition. St000871The number of very big ascents of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000929The constant term of the character polynomial of an integer partition. St000260The radius of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000259The diameter of a connected graph. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000137The Grundy value of an integer partition. St000145The Dyson rank of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000624The normalized sum of the minimal distances to a greater element. St000663The number of right floats of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000307The number of rowmotion orbits of a poset. St000646The number of big ascents of a permutation. St000837The number of ascents of distance 2 of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000664The number of right ropes of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001323The independence gap of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001975The corank of the alternating sign matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001871The number of triconnected components of a graph. St000534The number of 2-rises of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000640The rank of the largest boolean interval in a poset. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001570The minimal number of edges to add to make a graph Hamiltonian. St000782The indicator function of whether a given perfect matching is an L & P matching. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. 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 $eA$. St000456The monochromatic index of a connected graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000872The number of very big descents of a permutation. 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. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001665The number of pure excedances of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001960The number of descents of a permutation minus one if its first entry is not one. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000031The number of cycles in the cycle decomposition of a permutation. 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. St000761The number of ascents in an integer composition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000764The number of strong records in an integer composition.
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