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Your data matches 529 different statistics following compositions of up to 3 maps.
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Matching statistic: St001202
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
St001202: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra.
The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.
Matching statistic: St001230
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001290
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
Matching statistic: St000883
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 1
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000906
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000906: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> ([],2)
=> 1
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> ([],3)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St001184
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001210
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
Description
Gives 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.
Matching statistic: St000576
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000576: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000576: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> 1 = 2 - 1
[1,1,0,0]
=> {{1,2}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 0 = 1 - 1
Description
The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block and $j$ is the minimal element of a block.
Matching statistic: St000648
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0 = 1 - 1
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000683
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
The following 519 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001061The number of indices that are both descents and recoils of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000068The number of minimal elements in a poset. St000153The number of adjacent cycles of a permutation. St000273The domination number of a graph. St000306The bounce count of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000542The number of left-to-right-minima of a permutation. St000633The size of the automorphism group of a poset. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000916The packing number of a graph. St000990The first ascent of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000352The Elizalde-Pak rank of a permutation. St000366The number of double descents of a permutation. St000439The position of the first down step of a Dyck path. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000502The number of successions of a set partitions. St000534The number of 2-rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000731The number of double exceedences of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. 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) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000989The number of final rises of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001395The number of strictly unfriendly partitions of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001484The number of singletons of an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000015The number of peaks 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. St000053The number of valleys of the Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000504The cardinality of the first block of a set partition. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000823The number of unsplittable factors of the set partition. St000843The decomposition number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000920The logarithmic height of a Dyck path. St000925The number of topologically connected components of a set partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001050The number of terminal closers of a set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001621The number of atoms of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. 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. St001732The number of peaks visible from the left. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000013The height of a Dyck path. St000022The number of fixed points of a permutation. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000203The number of external nodes of a binary tree. St000234The number of global ascents of a permutation. St000248The number of anti-singletons of a set partition. St000331The number of upper interactions of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000381The largest part of an integer composition. St000386The number of factors DDU in a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000475The number of parts equal to 1 in a partition. St000496The rcs statistic of a set partition. St000528The height of a poset. St000546The number of global descents of a permutation. 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. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are 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. St000658The number of rises of length 2 of a Dyck path. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000738The first entry in the last row of a standard tableau. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000981The length of the longest zigzag subpath. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001092The number of distinct even parts of a partition. St001125The number of simple modules that satisfy the 2-regular condition 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. St001172The number of 1-rises at odd height of a Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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. St001343The dimension of the reduced incidence algebra of a poset. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001587Half of the largest even part of an integer partition. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001657The number of twos in an integer partition. St001717The largest size of an interval in a poset. St001718The number of non-empty open intervals in a poset. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001826The maximal number of leaves on a vertex of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000367The number of simsun double descents of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St000080The rank of the poset. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000694The number of affine bounded permutations that project to a given permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001225The vector space dimension of the first extension group between J and itself when J is 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001570The minimal number of edges to add to make a graph Hamiltonian. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000993The multiplicity of the largest part of an integer partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000650The number of 3-rises of a permutation. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000260The radius of a connected graph. St000649The number of 3-excedences of a permutation. St000456The monochromatic index of a connected graph. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. 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. St001904The length of the initial strictly increasing segment of a parking function. St000236The number of cyclical small weak excedances. St000664The number of right ropes of a permutation. St000982The length of the longest constant subword. 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 $eA$. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001960The number of descents of a permutation minus one if its first entry is not one. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001624The breadth of a lattice. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000929The constant term of the character polynomial of an integer partition. St001175The size of a partition minus the hook length of the base cell. 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 $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000862The number of parts of the shifted shape of a permutation. St000356The number of occurrences of the pattern 13-2. St000264The girth of a graph, which is not a tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001096The size of the overlap set of a permutation. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001586The number of odd parts smaller than the largest even part in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000701The protection number of a binary tree. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001955The number of natural descents for set-valued two row standard Young tableaux. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001867The number of alignments of type EN of a signed permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000806The semiperimeter of the associated bargraph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000074The number of special entries. St000100The number of linear extensions of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000287The number of connected components of a graph. St000307The number of rowmotion orbits of a poset. St000464The Schultz index of a connected graph. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000553The number of blocks of a graph. St000570The Edelman-Greene number of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001282The number of graphs with the same chromatic polynomial. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001500The global dimension of magnitude 1 Nakayama algebras. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001545The second Elser number of a connected graph. 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. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000258The burning number of a graph. St000284The Plancherel distribution on integer partitions. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000618The number of self-evacuating tableaux of given shape. St000646The number of big ascents of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000839The largest opener of a set partition. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001115The number of even descents of a permutation. St001118The acyclic chromatic index of a graph. St001128The exponens consonantiae of a partition. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001513The number of nested exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001763The Hurwitz number of an integer partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001913The number of preimages 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. 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. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000230Sum of the minimal elements of the blocks of a set partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000735The last entry on the main diagonal of a standard tableau. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000137The Grundy value of an integer partition. St000422The energy of a graph, if it is integral. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000934The 2-degree of an integer partition. 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. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001939The number of parts that are equal to their multiplicity in the integer partition. St001651The Frankl number of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001812The biclique partition number of a graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000699The toughness times the least common multiple of 1,. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves 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. St001561The value of the elementary symmetric function evaluated at 1. St001618The cardinality of the Frattini sublattice of a lattice. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.
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