Your data matches 469 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000791
(load all 24 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[]
 => [1,0]
 => [1,1,0,0]
 => 0
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000218
(load all 3 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[]
 => [] => [] => [] => 0
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000709
(load all 27 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,2,4,3,6,5] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,2,4,6,3,5] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,2,4,3,5,6] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,2,4,5,3,6] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,2,4,5,6,3] => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,2,5,3,6,4] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,2,5,6,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,2,5,3,4,6] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,2,3,5,4,6] => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,2,3,5,6,4] => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5] => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,2,3,6,4,5] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,2,3,4,6,5] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0
[]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0
Description
The number of occurrences of 14-2-3 or 14-3-2. The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
Matching statistic: St001193
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001193: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[]
 => [1,0]
 => [1] => [1,0]
 => 0
Description
The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module.
Matching statistic: St000425
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [3,1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [3,1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0
[]
 => [] => .
 => ? => ? = 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000428
(load all 5 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[]
 => [] => [] => [] => ? = 0
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
Matching statistic: St000560
(load all 2 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000560: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => {{1,2}}
 => {{1,2}}
 => 0
[1,0,1,0]
 => [3,1,2] => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
[1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => {{1,2,3}}
 => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => {{1},{2,3,4}}
 => 0
[1,1,0,0,1,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => {{1},{2},{3,4,5}}
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => {{1,5},{2,3},{4}}
 => 0
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => {{1,4},{2},{3,5}}
 => 0
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => 0
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => {{1},{2,4},{3,5}}
 => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => {{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => 0
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 3
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => {{1,3,4},{2,5}}
 => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 0
[]
 => [1] => {{1}}
 => {{1}}
 => ? = 0
Description
The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.
Matching statistic: St001735
(load all 3 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001735: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => [4,2,3,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[]
 => [] => [] => [] => ? = 0 + 1
Description
The number of permutations with the same set of runs. For example, the set of runs of $4132$ is $\{(13), (2), (4)\}$. The only other permutation with this set of runs is $4213$, so the statistic equals $2$ for these two permutations.
Matching statistic: St001786
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[]
 => [] => .
 => ?
 => ? = 0 + 1
Description
The 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. Alternatively, remark that the monomials of the polynomial $\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with $n$ east steps: the exponent of $z_i$ is the number of north steps before the $i$-th east step, see [2]. Thus, this statistic records the coefficients of the monomials. A formula for the coefficient of $z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]: $$ c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}. $$ This polynomial arises in a partial symmetrization process as follows, see [1]. For $w\in\frak{S}_n$, let $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Furthermore, let $$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$ Then $\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$.
Matching statistic: St000347
(load all 2 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 0
[1,0,1,0]
 => [1,2] => [1,2] => 0 => 0
[1,1,0,0]
 => [2,1] => [2,1] => 1 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 01 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 01 => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 01 => 0
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 11 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 011 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 011 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 011 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 011 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 011 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 101 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 010 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 011 => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 011 => 0
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 110 => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 101 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => 011 => 0
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 111 => 0
[]
 => [] => [] => ? => ? = 0
Description
The inversion sum of a binary word. A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
The following 459 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000348The non-inversion sum of a binary word. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St000022The number of fixed points of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000534The number of 2-rises of a permutation. St000068The number of minimal elements in a poset. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000563The number of overlapping pairs of blocks of a set partition. St001867The number of alignments of type EN of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000091The descent variation of a composition. St000562The number of internal points of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000872The number of very big descents of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three 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 $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000254The nesting number of a set partition. St000456The monochromatic index of a connected graph. St000570The Edelman-Greene number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001162The minimum jump of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001344The neighbouring number of a permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001949The rigidity index of a graph. St000258The burning number of a graph. St000906The length of the shortest maximal chain in a poset. St001352The number of internal nodes in the modular decomposition of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000649The number of 3-excedences of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000929The constant term of the character polynomial of an integer partition. St001082The number of boxed occurrences of 123 in a permutation. St001099The 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 leaf labelled binary trees. St001100The 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 leaf labelled trees. 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. 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. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001389The number of partitions of the same length below the given integer partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001527The cyclic permutation representation number of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. St001943The sum of the squares of the hook lengths of an integer partition. St000893The number of distinct diagonal sums of an alternating sign matrix. 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 $eA$. St001330The hat guessing number of a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000133The "bounce" of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000731The number of double exceedences of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. 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. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001524The degree of symmetry of a binary word. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001846The number of elements which do not have a complement in the lattice. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000527The width of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000909The number of maximal chains of maximal size in a poset. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000991The number of right-to-left minima of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001518The number of graphs with the same ordinary spectrum as the given graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001729The number of visible descents of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001820The size of the image of the pop stack sorting operator. St001928The number of non-overlapping descents in a permutation. St000058The order of a permutation. St000084The number of subtrees. St000105The number of blocks in the set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000401The size of the symmetry class of a permutation. St000417The size of the automorphism group of the ordered tree. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000823The number of unsplittable factors of the set partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001569The maximal modular displacement of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000315The number of isolated vertices of a graph. St000632The jump number of the poset. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001429The number of negative entries in a signed permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000096The number of spanning trees of a graph. St000287The number of connected components of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000309The number of vertices with even degree. St000450The number of edges minus the number of vertices plus 2 of a graph. St000739The first entry in the last row of a semistandard tableau. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000958The number of Bruhat factorizations of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001260The permanent of an alternating sign matrix. St001410The minimal entry of a semistandard tableau. St001498The normalised height of a Nakayama algebra with magnitude 1. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours 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. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001828The Euler characteristic of a graph. St001946The number of descents in a parking function. St000134The size of the orbit of an alternating sign matrix under gyration. St000822The Hadwiger number of the graph. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001060The distinguishing index of a graph. 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 $eA$. 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 $eA$. St001555The order of a signed permutation. St001734The lettericity of a graph. St001893The flag descent of a signed permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph.