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Your data matches 691 different statistics following compositions of up to 3 maps.
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Matching statistic: St001135
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> 2
[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
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001201
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> 2
[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
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St000007
(load all 121 compositions to match this statistic)
(load all 121 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[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] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 2
[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 saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000011
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[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]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
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.
Matching statistic: St000025
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [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]
=> 1
[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]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[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]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[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
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000335
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [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]
=> 1
[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]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[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]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[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
The difference of lower and upper interactions.
An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Matching statistic: St000678
(load all 59 compositions to match this statistic)
(load all 59 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001024
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001050
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[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 terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St001184
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ 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
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[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]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[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.
The following 681 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of 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 $c_{n-1}$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001484The number of singletons of an integer partition. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000546The number of global descents of a permutation. 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. St000759The smallest missing part in an integer partition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000883The number of longest increasing subsequences of a permutation. St000971The smallest closer of a set partition. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001316The domatic number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001461The number of topologically connected components of the chord diagram of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000133The "bounce" of a permutation. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000352The Elizalde-Pak rank of a permutation. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000486The number of cycles of length at least 3 of a permutation. 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. 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. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000731The number of double exceedences of a permutation. St000738The first entry in the last row of a standard tableau. 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. 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. St001086The number of occurrences of the consecutive pattern 132 in a permutation. 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000381The largest 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. St000501The size of the first part in the decomposition of a permutation. 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. St000662The staircase size of the code of a permutation. St000675The number of centered multitunnels of a Dyck path. St000701The protection number of a binary tree. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000906The length of the shortest maximal chain in a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000916The packing number of a graph. 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. St001029The size of the core of a graph. St001052The length of the exterior of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001116The game chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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:
St001261The Castelnuovo-Mumford regularity of a graph. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001481The minimal height of a peak of a Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001621The number of atoms of a lattice. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001829The common independence number of a graph. St001933The largest multiplicity of a part in an integer partition. St001963The tree-depth of a graph. St000013The height of a Dyck path. St000022The number of fixed points of a permutation. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000120The number of left tunnels of a Dyck path. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000203The number of external nodes of a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000331The number of upper interactions of a Dyck path. St000356The number of occurrences of the pattern 13-2. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. 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. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000481The number of upper covers of a partition in dominance order. St000502The number of successions of a set partitions. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000534The number of 2-rises of a permutation. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St000654The first descent of a permutation. St000658The number of rises of length 2 of a Dyck path. St000674The number of hills of a Dyck path. St000683The 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000734The last entry in the first 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. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000918The 2-limited packing number of a graph. St000947The major index east count of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000981The length of the longest zigzag subpath. St000985The number of positive eigenvalues of the adjacency matrix of the graph. 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. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001062The maximal size of a block of a set partition. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001133The smallest label in the subtree rooted at the sister of 1 in 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. 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. 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. St001252Half the sum of the even parts of a partition. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001394The genus of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. 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. St001552The number of inversions between excedances and fixed points of a permutation. St001587Half of the largest even part of an integer partition. 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. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001728The number of invisible descents of a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. 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. St000061The number of nodes on the left branch of a binary tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000633The size of the automorphism group of a poset. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. 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. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000181The number of connected components of the Hasse diagram for the poset. 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. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000823The number of unsplittable factors of the set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000914The sum of the values of the Möbius function of a poset. St000924The number of topologically connected components of a perfect matching. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000221The number of strong fixed points of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000461The rix statistic of a permutation. St000498The lcs statistic of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000753The Grundy value for the game of Kayles on a binary word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic 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. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001592The maximal number of simple paths between any two different vertices of a graph. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000756The sum of the positions of the left to right maxima of a permutation. St001948The number of augmented double ascents of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000219The number of occurrences of the pattern 231 in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000768The number of peaks in an integer composition. St000741The Colin de Verdière graph invariant. St001889The size of the connectivity set of a signed permutation. St001549The number of restricted non-inversions between exceedances. 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. St000942The number of critical left to right maxima of the parking functions. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001737The number of descents of type 2 in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000929The constant term of the character polynomial of an integer partition. St000982The length of the longest constant subword. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. 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 $eA$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001557The number of inversions of the second entry of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001864The number of excedances of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000871The number of very big ascents of a permutation. St000907The number of maximal antichains of minimal length in a poset. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000264The girth of a graph, which is not a tree. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000492The rob statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001151The number of blocks with odd minimum. 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 $eA$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001569The maximal modular displacement of a permutation. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000091The descent variation of a composition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000552The number of cut vertices of a graph. 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. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000806The semiperimeter of the associated bargraph. St000839The largest opener of a set partition. St000944The 3-degree of an integer partition. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001781The interlacing number of a set partition. St001867The number of alignments of type EN of a signed permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000230Sum of the minimal elements of the blocks of a set partition. St000456The monochromatic index of a connected graph. St000422The energy of a graph, if it is integral. St000145The Dyson rank of a partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. 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. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000284The Plancherel distribution on integer partitions. St000477The weight of a partition according to Alladi. 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. 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. St001060The distinguishing index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000023The number of inner peaks of a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000253The crossing number of a set partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000570The Edelman-Greene number of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000736The last entry in the first row of a semistandard tableau. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000988The orbit size of a permutation under Foata's bijection. St000997The even-odd crank of an integer partition. 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001115The number of even descents of a permutation. St001162The minimum jump of a permutation. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001487The number of inner corners of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. 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. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000075The orbit size of a standard tableau under promotion. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000166The depth minus 1 of an ordered tree. St000177The number of free tiles in the pattern. St000178Number of free entries. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000232The number of crossings of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000299The number of nonisomorphic vertex-induced subtrees. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000516The number of stretching pairs of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. 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. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000646The number of big ascents of a permutation. St000664The number of right ropes of a permutation. St000682The Grundy value of Welter's game on a binary word. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000842The breadth of a permutation. St000928The sum of the coefficients of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of 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. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. 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. St001377The major index minus the number of inversions of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001623The number of doubly irreducible elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001637The number of (upper) dissectors of a poset. St001651The Frankl number of a lattice. St001667The maximal size of a pair of weak twins for a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001684The reduced word complexity of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001822The number of alignments of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001911A descent variant minus the number of inversions. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000094The depth of an ordered tree. St000521The number of distinct subtrees of an ordered tree. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. 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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