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Your data matches 577 different statistics following compositions of up to 3 maps.
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Matching statistic: St001059
(load all 160 compositions to match this statistic)
(load all 160 compositions to match this statistic)
St001059: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 0 = 1 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => 0 = 1 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => 0 = 1 - 1
Description
Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation.
Matching statistic: St001550
(load all 168 compositions to match this statistic)
(load all 168 compositions to match this statistic)
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 0 = 1 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => 0 = 1 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => 0 = 1 - 1
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001551
(load all 189 compositions to match this statistic)
(load all 189 compositions to match this statistic)
St001551: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 0 = 1 - 1
[2,1,3] => 0 = 1 - 1
[2,3,1] => 0 = 1 - 1
[3,1,2] => 0 = 1 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => 0 = 1 - 1
[1,3,4,2] => 0 = 1 - 1
[1,4,2,3] => 0 = 1 - 1
[1,4,3,2] => 0 = 1 - 1
[2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => 0 = 1 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 0 = 1 - 1
[3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => 0 = 1 - 1
[4,1,2,3] => 0 = 1 - 1
[4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 0 = 1 - 1
[1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => 0 = 1 - 1
[1,2,5,3,4] => 0 = 1 - 1
[1,2,5,4,3] => 0 = 1 - 1
[1,3,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => 0 = 1 - 1
[1,4,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => 0 = 1 - 1
[1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => 0 = 1 - 1
Description
The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St000036
(load all 62 compositions to match this statistic)
(load all 62 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [2,1,4,3] => 1
[3,4,1,2] => [2,4,1,3] => 1
[3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,3,2,5,4] => 1
[1,4,5,2,3] => [1,3,5,2,4] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St001208
(load all 324 compositions to match this statistic)
(load all 324 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The 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 $K[x]/(x^n)$.
Matching statistic: St001518
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001518: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([],2)
=> 1
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 1
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([],4)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => ([],5)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
Description
The number of graphs with the same ordinary spectrum as the given graph.
Matching statistic: St000132
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000132: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 0 = 1 - 1
[1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 0 = 1 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0 = 1 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 0 = 1 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 0 = 1 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A159773]] counts binary trees avoiding this pattern.
Matching statistic: St000375
(load all 72 compositions to match this statistic)
(load all 72 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,3,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0 = 1 - 1
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000404
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,3,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0 = 1 - 1
Description
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].
Matching statistic: St000407
(load all 299 compositions to match this statistic)
(load all 299 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000407: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000407: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,3,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => 0 = 1 - 1
Description
The number of occurrences of the pattern 2143 in a permutation.
A permutation $\pi$ avoids this pattern if and only if it is ''vexillary'' as introduced in [1].
The following 567 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000666The number of right tethers of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001272The number of graphs with the same degree sequence. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000370The genus of a graph. 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 $3$. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 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. St000447The number of pairs of vertices of a graph with distance 3. St000664The number of right ropes of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000807The sum of the heights of the valleys of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001271The competition number of a graph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001394The genus of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". 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. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. 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. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000183The side length of the Durfee square of an integer partition. St000273The domination number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000883The number of longest increasing subsequences of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001393The induced matching number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001624The breadth of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001716The 1-improper chromatic number of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001957The number of Hasse diagrams with a given underlying undirected graph. St000002The number of occurrences of the pattern 123 in a permutation. St000017The number of inversions of a standard tableau. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000150The floored half-sum of the multiplicities of a partition. 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. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000257The number of distinct parts of a partition that occur at least twice. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000546The number of global descents of a permutation. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000630The length of the shortest palindromic decomposition of a binary word. St000671The maximin edge-connectivity for choosing a subgraph. St000731The number of double exceedences of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000871The number of very big ascents of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension 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. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. 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. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. 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. 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001377The major index minus the number of inversions of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001471The magnitude of a Dyck path. St001535The number of cyclic alignments of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001638The book thickness of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001781The interlacing number of a set partition. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001810The number of fixed points of a permutation smaller than its largest moved point. 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. St001871The number of triconnected components of a graph. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000768The number of peaks in an integer composition. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000842The breadth of a permutation. St000872The number of very big descents of a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001552The number of inversions between excedances and fixed points of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St000047The number of standard immaculate tableaux of a given shape. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000392The length of the longest run of ones in a binary word. St000640The rank of the largest boolean interval in a poset. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000990The first ascent of a permutation. St001220The width of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000091The descent variation of a composition. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000562The number of internal points of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 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. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000658The number of rises of length 2 of a Dyck path. St000710The number of big deficiencies of a permutation. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001520The number of strict 3-descents. St001783The number of odd automorphisms of a graph. St001964The interval resolution global dimension of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000456The monochromatic index of a connected graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001613The binary logarithm of the size of the center of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St000914The sum of the values of the Möbius function of a poset. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001555The order of a signed permutation. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001626The number of maximal proper sublattices of a lattice. St000553The number of blocks of a graph. St000916The packing number of a graph. 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. St001739The number of graphs with the same edge polytope as the given graph. 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. St000629The defect of a binary word. St000878The number of ones minus the number of zeros of a binary word. 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 $eA$. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000889The number of alternating sign matrices with the same antidiagonal sums. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000488The number of cycles of a permutation of length at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001741The largest integer such that all patterns of this size are contained in the permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St001763The Hurwitz number of an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000159The number of distinct parts of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000287The number of connected components of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000783The side length of the largest staircase partition fitting into a partition. St000897The number of different multiplicities of parts of an integer partition. St000935The number of ordered refinements of an integer partition. 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,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001129The product of the squares of the parts of a partition. 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 $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001765The number of connected components of the friends and strangers graph. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000735The last entry on the main diagonal of a standard tableau. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000478Another weight of a partition according to Alladi. St000759The smallest missing part in an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001429The number of negative entries in a signed permutation. St001933The largest multiplicity of a part in an integer partition. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000475The number of parts equal to 1 in a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001889The size of the connectivity set of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001621The number of atoms of a lattice. St001330The hat guessing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000297The number of leading ones in a binary word. St000022The number of fixed points of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001863The number of weak excedances of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001430The number of positive entries in a signed permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001316The domatic number of a graph. St001768The number of reduced words of a signed permutation. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001323The independence gap of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001363The Euler characteristic of a graph according to Knill. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001884The number of borders of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000074The number of special entries. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000552The number of cut vertices of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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 $K[x]/(x^n)$. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001927Sparre Andersen's number of positives of a signed permutation. St001052The length of the exterior of a permutation. St001410The minimal entry of a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001857The number of edges in the reduced word graph of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000264The girth of a graph, which is not a tree. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. St000286The number of connected components of the complement of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001570The minimal number of edges to add to make a graph Hamiltonian. St000907The number of maximal antichains of minimal length in a poset. St001347The number of pairs of vertices of a graph having the same neighbourhood. St000097The order of the largest clique of the graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000917The open packing number of a graph. St000948The chromatic discriminant of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001654The monophonic hull number of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000981The length of the longest zigzag subpath.
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