Processing math: 100%

Your data matches 859 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,1,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,1,1]
=> 6
[3,1,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,1,1]
=> 7
[4,1,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
Description
The size of a partition. This statistic is the constant statistic of the level sets.
St000459: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,1,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,1,1]
=> 6
[3,1,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,1,1]
=> 7
[4,1,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
St000460: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,1,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,1,1]
=> 6
[3,1,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,1,1]
=> 7
[4,1,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
Description
The hook length of the last cell along the main diagonal of an integer partition.
St000870: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,1,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,1,1]
=> 6
[3,1,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,1,1]
=> 7
[4,1,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells (i,i) of a partition.
St001382: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 2 = 3 - 1
[2,1]
=> 2 = 3 - 1
[1,1,1]
=> 2 = 3 - 1
[4]
=> 3 = 4 - 1
[3,1]
=> 3 = 4 - 1
[2,1,1]
=> 3 = 4 - 1
[1,1,1,1]
=> 3 = 4 - 1
[5]
=> 4 = 5 - 1
[4,1]
=> 4 = 5 - 1
[3,1,1]
=> 4 = 5 - 1
[2,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1]
=> 4 = 5 - 1
[6]
=> 5 = 6 - 1
[5,1]
=> 5 = 6 - 1
[4,1,1]
=> 5 = 6 - 1
[3,1,1,1]
=> 5 = 6 - 1
[2,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> 5 = 6 - 1
[7]
=> 6 = 7 - 1
[6,1]
=> 6 = 7 - 1
[5,1,1]
=> 6 = 7 - 1
[4,1,1,1]
=> 6 = 7 - 1
[3,1,1,1,1]
=> 6 = 7 - 1
[2,1,1,1,1,1]
=> 6 = 7 - 1
[1,1,1,1,1,1,1]
=> 6 = 7 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 1000 => 3
[2,1]
=> 1010 => 3
[1,1,1]
=> 1110 => 3
[4]
=> 10000 => 4
[3,1]
=> 10010 => 4
[2,1,1]
=> 10110 => 4
[1,1,1,1]
=> 11110 => 4
[5]
=> 100000 => 5
[4,1]
=> 100010 => 5
[3,1,1]
=> 100110 => 5
[2,1,1,1]
=> 101110 => 5
[1,1,1,1,1]
=> 111110 => 5
[6]
=> 1000000 => 6
[5,1]
=> 1000010 => 6
[4,1,1]
=> 1000110 => 6
[3,1,1,1]
=> 1001110 => 6
[2,1,1,1,1]
=> 1011110 => 6
[1,1,1,1,1,1]
=> 1111110 => 6
[7]
=> 10000000 => 7
[6,1]
=> 10000010 => 7
[5,1,1]
=> 10000110 => 7
[4,1,1,1]
=> 10001110 => 7
[3,1,1,1,1]
=> 10011110 => 7
[2,1,1,1,1,1]
=> 10111110 => 7
[1,1,1,1,1,1,1]
=> 11111110 => 7
Description
The number of inversions of a binary word.
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 6
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 7
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00095: Integer partitions to binary wordBinary words
St000921: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 1000 => 2 = 3 - 1
[2,1]
=> 1010 => 2 = 3 - 1
[1,1,1]
=> 1110 => 2 = 3 - 1
[4]
=> 10000 => 3 = 4 - 1
[3,1]
=> 10010 => 3 = 4 - 1
[2,1,1]
=> 10110 => 3 = 4 - 1
[1,1,1,1]
=> 11110 => 3 = 4 - 1
[5]
=> 100000 => 4 = 5 - 1
[4,1]
=> 100010 => 4 = 5 - 1
[3,1,1]
=> 100110 => 4 = 5 - 1
[2,1,1,1]
=> 101110 => 4 = 5 - 1
[1,1,1,1,1]
=> 111110 => 4 = 5 - 1
[6]
=> 1000000 => 5 = 6 - 1
[5,1]
=> 1000010 => 5 = 6 - 1
[4,1,1]
=> 1000110 => 5 = 6 - 1
[3,1,1,1]
=> 1001110 => 5 = 6 - 1
[2,1,1,1,1]
=> 1011110 => 5 = 6 - 1
[1,1,1,1,1,1]
=> 1111110 => 5 = 6 - 1
[7]
=> 10000000 => 6 = 7 - 1
[6,1]
=> 10000010 => 6 = 7 - 1
[5,1,1]
=> 10000110 => 6 = 7 - 1
[4,1,1,1]
=> 10001110 => 6 = 7 - 1
[3,1,1,1,1]
=> 10011110 => 6 = 7 - 1
[2,1,1,1,1,1]
=> 10111110 => 6 = 7 - 1
[1,1,1,1,1,1,1]
=> 11111110 => 6 = 7 - 1
Description
The number of internal inversions of a binary word. Let ˉw be the non-decreasing rearrangement of w, that is, ˉw is sorted. An internal inversion is a pair i<j such that wi>wj and ˉwi=ˉwj. For example, the word 110 has two inversions, but only the second is internal.
Mp00043: Integer partitions to Dyck pathDyck paths
St001643: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> [1,1,1,0,0,0,1,0]
=> 5 = 3 + 2
[2,1]
=> [1,0,1,0,1,0]
=> 5 = 3 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 5 = 3 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 4 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 6 = 4 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 6 = 4 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 4 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 7 = 5 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 7 = 5 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 7 = 5 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7 = 5 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7 = 5 + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 8 = 6 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 8 = 6 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 8 = 6 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 8 = 6 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 8 = 6 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 8 = 6 + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 9 = 7 + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 9 = 7 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 9 = 7 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 9 = 7 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 9 = 7 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 9 = 7 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 9 = 7 + 2
Description
The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000011
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00103: Dyck paths peeling mapDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
The following 849 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000018The number of inversions of a permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000528The height of a poset. St000657The smallest part of an integer composition. St000674The number of hills of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000808The number of up steps of the associated bargraph. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn1] such that n=c0<ci for all i>0 a Dyck path as follows: St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001485The modular major index of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000053The number of valleys of the Dyck path. St000070The number of antichains in a poset. St000385The number of vertices with out-degree 1 in a binary tree. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000806The semiperimeter of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn1] by adding c0 to cn1. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001631The number of simple modules S with dimExt1(S,A)=1 in the incidence algebra A of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. 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. St001955The number of natural descents for set-valued two row standard Young tableaux. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000013The height of a Dyck path. St000022The number of fixed points of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000184The size of the centralizer of any permutation of given cycle type. St000203The number of external nodes of a binary tree. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000294The number of distinct factors of a binary word. St000308The height of the tree associated to a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000461The rix statistic of a permutation. St000469The distinguishing number of a graph. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000518The number of distinct subsequences in a binary word. St000520The number of patterns in a permutation. St000529The number of permutations whose descent word is the given binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000625The sum of the minimal distances to a greater element. St000636The hull number of a graph. St000654The first descent of a permutation. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000770The major index of an integer partition when read from bottom to top. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000863The length of the first row of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000926The clique-coclique number of a graph. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000979Half of MacMahon's equal index of a Dyck path. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001074The number of inversions of the cyclic embedding of a permutation. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001439The number of even weak deficiencies and of odd weak exceedences. St001441The number of non-empty connected induced subgraphs of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001497The position of the largest weak excedence of a permutation. St001527The cyclic permutation representation number of an integer partition. St001566The length of the longest arithmetic progression in a permutation. St001571The Cartan determinant of the integer partition. St001614The cyclic permutation representation number of a skew partition. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001838The number of nonempty primitive factors of a binary word. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001910The height of the middle non-run of a Dyck path. St000003The number of standard Young tableaux of the partition. St000007The number of saliances of the permutation. St000010The length of the partition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000054The first entry of the permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000064The number of one-box pattern of a permutation. St000081The number of edges of a graph. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000215The number of adjacencies of a permutation, zero appended. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000296The length of the symmetric border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000325The width of the tree associated to a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000507The number of ascents of a standard tableau. St000519The largest length of a factor maximising the subword complexity. St000532The total number of rook placements on a Ferrers board. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000627The exponent of a binary word. St000672The number of minimal elements in Bruhat order not less than the permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000691The number of changes of a binary word. St000696The number of cycles in the breakpoint graph of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000778The metric dimension of a graph. St000825The sum of the major and the inverse major index of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000874The position of the last double rise in a Dyck path. St000883The number of longest increasing subsequences of a permutation. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000984The number of boxes below precisely one peak. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001479The number of bridges of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001541The Gini index of an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001658The total number of rook placements on a Ferrers board. St001777The number of weak descents in an integer composition. St001780The order of promotion on the set of standard tableaux of given shape. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001917The order of toric promotion on the set of labellings of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000060The greater neighbor of the maximum. St000088The row sums of the character table of the symmetric group. St000141The maximum drop size of a permutation. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000209Maximum difference of elements in cycles. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000214The number of adjacencies of a permutation. St000238The number of indices that are not small weak excedances. St000295The length of the border of a binary word. St000313The number of degree 2 vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000377The dinv defect of an integer partition. St000448The number of pairs of vertices of a graph with distance 2. St000475The number of parts equal to 1 in a partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000619The number of cyclic descents of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000653The last descent of a permutation. St000662The staircase size of the code of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001127The sum of the squares of the parts of a partition. St001130The number of two successive successions in a permutation. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001246The maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001308The number of induced paths on three vertices in a graph. St001489The maximum of the number of descents and the number of inverse descents. St001521Half the total irregularity of a graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001933The largest multiplicity of a part in an integer partition. St001958The degree of the polynomial interpolating the values of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000836The number of descents of distance 2 of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001306The number of induced paths on four vertices in a graph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St001759The Rajchgot index of a permutation. St000973The length of the boundary of an ordered tree. St000050The depth or height of a binary tree. St001094The depth index of a set partition. St001161The major index north count of a Dyck path. St000915The Ore degree of a graph. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001622The number of join-irreducible elements of a lattice. St000719The number of alignments in a perfect matching. St001397Number of pairs of incomparable elements in a finite poset. St001430The number of positive entries in a signed permutation. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000004The major index of a permutation. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000224The sorting index of a permutation. St000332The positive inversions of an alternating sign matrix. St000339The maf index of a permutation. St000446The disorder of a permutation. St001428The number of B-inversions of a signed permutation. St000029The depth of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001278The number of indecomposable modules that are fixed by τΩ1 composed with its inverse in the corresponding Nakayama algebra. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000015The number of peaks of a Dyck path. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000954Number of times the corresponding LNakayama algebra has Exti(D(A),A)=0 for i>0. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn1] by adding c0 to cn1. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St000028The number of stack-sorts needed to sort a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000117The number of centered tunnels of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000309The number of vertices with even degree. St000314The number of left-to-right-maxima of a permutation. St000315The number of isolated vertices of a graph. St000335The difference of lower and upper interactions. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000443The number of long tunnels of a Dyck path. St000471The sum of the ascent tops of a permutation. St000503The maximal difference between two elements in a common block. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000733The row containing the largest entry of a standard tableau. St000890The number of nonzero entries in an alternating sign matrix. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules S with ExtiA(S,A)=0 for all i=0,1,...,g1 in the corresponding Nakayama algebra A with global dimension g. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001481The minimal height of a peak of a Dyck path. St001519The pinnacle sum of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001828The Euler characteristic of a graph. St001959The product of the heights of the peaks of a Dyck path. St000058The order of a permutation. St000094The depth of an ordered tree. St000167The number of leaves of an ordered tree. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000291The number of descents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000451The length of the longest pattern of the form k 1 2. St000505The biggest entry in the block containing the 1. St000530The number of permutations with the same descent word as the given permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001516The number of cyclic bonds of a permutation. St001649The length of a longest trail in a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000083The number of left oriented leafs of a binary tree except the first one. St000389The number of runs of ones of odd length in a binary word. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. 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. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000924The number of topologically connected components of a perfect matching. St001345The Hamming dimension of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001401The number of distinct entries in a semistandard tableau. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001726The number of visible inversions of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000327The number of cover relations in a poset. St000673The number of non-fixed points of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001468The smallest fixpoint of a permutation. St000051The size of the left subtree of a binary tree. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000780The size of the orbit under rotation of a perfect matching. St001077The prefix exchange distance of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000163The size of the orbit of the set partition under rotation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. 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. St001268The size of the largest ordinal summand in the poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001834The number of non-isomorphic minors of a graph. St001948The number of augmented double ascents of a permutation. 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]/(xn). St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000945The number of matchings in the dihedral orbit of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000135The number of lucky cars of the parking function. St001409The maximal entry of a semistandard tableau. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001927Sparre Andersen's number of positives of a signed permutation. St000017The number of inversions of a standard tableau. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001424The number of distinct squares in a binary word. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000735The last entry on the main diagonal of a standard tableau. St001935The number of ascents in a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000071The number of maximal chains in a poset. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000522The number of 1-protected nodes of a rooted tree. St000356The number of occurrences of the pattern 13-2. St000521The number of distinct subtrees of an ordered tree. St001555The order of a signed permutation. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001964The interval resolution global dimension of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000075The orbit size of a standard tableau under promotion. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000493The los statistic of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000638The number of up-down runs of a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001731The factorization defect of a permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001926Sparre Andersen's position of the maximum of a signed permutation. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000116The major index of a semistandard tableau obtained by standardizing. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000253The crossing number of a set partition. St000328The maximum number of child nodes in a tree. St000358The number of occurrences of the pattern 31-2. St000539The number of odd inversions of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000779The tier of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001469The holeyness of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000023The number of inner peaks of a permutation. St000154The sum of the descent bottoms of a permutation. St000353The number of inner valleys of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000872The number of very big descents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001114The number of odd descents of a permutation. St001162The minimum jump of a permutation. 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]/(xn). St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. 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]/(xn). St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001413Half the length of the longest even length palindromic prefix of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001490The number of connected components of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001737The number of descents of type 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001839The number of excedances of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St000039The number of crossings of a permutation. St000091The descent variation of a composition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000156The Denert index 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. St000222The number of alignments in the permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in 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. St000494The number of inversions of distance at most 3 of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000540The sum of the entries of a parking function minus its length. 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. St000562The number of internal points of 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. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. 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. 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. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. 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. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000677The standardized bi-alternating inversion number 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. St000756The sum of the positions of the left to right maxima of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings 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. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records 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. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001807The lower middle entry of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000037The sign of a permutation. St001721The degree of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St001671Haglund's hag of a permutation. St000165The sum of the entries of a parking function. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St000690The size of the conjugacy class of a permutation. St000016The number of attacking pairs of a standard tableau. St001852The size of the conjugacy class of the signed permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001885The number of binary words with the same proper border set. St000324The shape of the tree associated to a permutation. St001528The number of permutations such that the product with the permutation has the same number of fixed points.