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Your data matches 648 different statistics following compositions of up to 3 maps.
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Matching statistic: St001289
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
St001289: 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
[1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
Description
The 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.
This n-fold tensor product seems to be always injective.
Matching statistic: St000004
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000154
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
Matching statistic: St000156
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
Matching statistic: St000305
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000334
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000334: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000334: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
The maz index, the major index of a permutation after replacing fixed points by zeros.
The descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
Matching statistic: St000344
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St000403
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000403: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000403: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 0
Description
The Szeged index minus the Wiener index of a graph.
It is known that the Szeged index is at least as much as the Wiener index. For $2$-connected graphs on $n$ vertices, the difference is at least $2n-6$.
For the two individual statistics see [[St000263]] and [[St000265]].
Matching statistic: St000463
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,3,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,3,1,2,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 0
Description
The number of admissible inversions of a permutation.
Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$.
An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions:
$1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
Matching statistic: St000692
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000692: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000692: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,2,3,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [1,2,3,4,5,6] => 0
Description
Babson and Steingrímsson's statistic of a permutation.
In terms of generalized patterns this is
$$
(13-2) + (21-3) + (32-1) + (21).
$$
Here, $(\pi)$ denotes the number of times the pattern $\pi$ occurs in a permutation, and letters in the pattern which are not separated by a dash must appear consecutively.
The following 638 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000828The spearman's rho of a permutation and the identity permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001402The number of separators in a permutation. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001902The number of potential covers of a poset. St000028The number of stack-sorts needed to sort a permutation. St000763The sum of the positions of the strong records of an integer composition. St001285The number of primes in the column sums of the two line notation of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001528The number of permutations such that the product with the permutation has the same number of fixed points. 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. St000346The number of coarsenings of a partition. St000008The major index of the composition. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000022The number of fixed points of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000141The maximum drop size of a permutation. St000143The largest repeated part of a partition. St000155The number of exceedances (also excedences) of a permutation. St000185The weighted size of a partition. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000224The sorting index of a permutation. St000225Difference between largest and smallest parts in a partition. St000235The number of indices that are not cyclical small weak excedances. St000237The number of small exceedances. St000268The number of strongly connected orientations of a graph. St000290The major index of a binary word. St000297The number of leading ones in a binary word. St000304The load of a permutation. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000316The number of non-left-to-right-maxima of a permutation. St000330The (standard) major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000339The maf index of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000350The sum of the vertex degrees of a graph. St000359The number of occurrences of the pattern 23-1. St000374The number of exclusive right-to-left minima of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000422The energy of a graph, if it is integral. St000446The disorder of a permutation. St000465The first Zagreb index of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000491The number of inversions of a set partition. St000493The los statistic of a set partition. St000496The rcs statistic of a set partition. St000499The rcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000565The major index of a set partition. St000571The F-index (or forgotten topological index) of a graph. St000572The dimension exponent of a set partition. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000648The number of 2-excedences of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000769The major index of a composition regarded as a word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000885The number of critical steps in the Catalan decomposition of a binary word. St000915The Ore degree of a graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St000947The major index east count of a Dyck path. St000995The largest even part of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001090The number of pop-stack-sorts needed to sort a permutation. St001094The depth index of a set partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001362The normalized Knill dimension of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001375The pancake length of a permutation. St001394The genus of a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001618The cardinality of the Frattini sublattice of a lattice. St001669The number of single rises in a Dyck path. 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. St001692The number of vertices with higher degree than the average degree in a graph. St001696The natural major index of a standard Young tableau. St001699The major index of a standard tableau minus the weighted size of its shape. St001708The number of pairs of vertices of different degree in a graph. St001726The number of visible inversions of a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001783The number of odd automorphisms of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001842The major index of a set partition. St001843The Z-index of a set partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000054The first entry of the permutation. St000058The order of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000163The size of the orbit of the set partition under rotation. St000182The number of permutations whose cycle type is the given integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000240The number of indices that are not small excedances. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000349The number of different adjacency matrices of a graph. St000378The diagonal inversion number of an integer partition. St000416The number of inequivalent increasing trees of an ordered tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000456The monochromatic index of a connected graph. St000501The size of the first part in the decomposition of a permutation. St000517The Kreweras number of an integer partition. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000638The number of up-down runs of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000722The number of different neighbourhoods in a graph. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000946The sum of the skew hook positions in a Dyck path. St000948The chromatic discriminant of a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001277The degeneracy of a graph. St001303The number of dominating sets of vertices of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001358The largest degree of a regular subgraph of a graph. St001386The number of prime labellings of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001500The global dimension of magnitude 1 Nakayama algebras. St001546The number of monomials in the Tutte polynomial of a graph. St001564The value of the forgotten symmetric functions when all variables set to 1. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001780The order of promotion on the set of standard tableaux of given shape. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001808The box weight or horizontal decoration of a Dyck path. St001850The number of Hecke atoms of a permutation. 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. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001931The weak major index of an integer composition regarded as a word. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000981The length of the longest zigzag subpath. St001029The size of the core of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001814The number of partitions interlacing the given partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000961The shifted major index of a permutation. St000653The last descent of a permutation. St000673The number of non-fixed points of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000830The total displacement 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. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000690The size of the conjugacy class of a permutation. St001838The number of nonempty primitive factors of a binary word. St000216The absolute length of a permutation. St000391The sum of the positions of the ones in a binary word. St000462The major index minus the number of excedences of a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000567The sum of the products of all pairs of parts. St000647The number of big descents of a permutation. St000693The modular (standard) major index of a standard tableau. St000792The Grundy value for the game of ruler on a binary word. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000936The number of even values of the symmetric group character corresponding to the partition. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001703The villainy of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000652The maximal difference between successive positions of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000844The size of the largest block in the direct sum decomposition of a permutation. St001128The exponens consonantiae of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. 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 $eA$. 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. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. 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. St001141The number of occurrences of hills of size 3 in a Dyck path. 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. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000096The number of spanning trees of a graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St000455The second largest eigenvalue of a graph if it is integral. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000219The number of occurrences of the pattern 231 in a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000542The number of left-to-right-minima of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000990The first ascent of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000260The radius of a connected graph. St000983The length of the longest alternating subword. St000929The constant term of the character polynomial of an integer partition. St000909The number of maximal chains of maximal size in a poset. St001060The distinguishing index of a graph. St000002The number of occurrences of the pattern 123 in a permutation. St000534The number of 2-rises of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000742The number of big ascents of a permutation after prepending zero. 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. St001718The number of non-empty open intervals in a poset. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000667The greatest common divisor of the parts of the partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001571The Cartan determinant of the integer partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000651The maximal size of a rise in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000877The depth of the binary word interpreted as a path. St000223The number of nestings in the 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 $3$. St000478Another weight of a partition according to Alladi. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St000360The number of occurrences of the pattern 32-1. St001411The number of patterns 321 or 3412 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000367The number of simsun double descents of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000913The number of ways to refine the partition into singletons. St000933The number of multipartitions of sizes given by 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. 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. 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 $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in 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. St000630The length of the shortest palindromic decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St001175The size of a partition minus the hook length of the base cell. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000218The number of occurrences of the pattern 213 in a permutation. St001964The interval resolution global dimension of a poset. St000255The number of reduced Kogan faces with the permutation as type. St000842The breadth of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001396Number of triples of incomparable elements in a finite poset. St000078The number of alternating sign matrices whose left key is the permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000355The number of occurrences of the pattern 21-3. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001381The fertility of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001081The number of minimal length factorizations of a permutation into star transpositions. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000095The number of triangles of a graph. St000357The number of occurrences of the pattern 12-3. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000358The number of occurrences of the pattern 31-2. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000461The rix statistic of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001095The number of non-isomorphic posets with precisely one further covering relation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001651The Frankl number of a lattice. St001684The reduced word complexity of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000037The sign of a permutation. St000081The number of edges of a graph. St000160The multiplicity of the smallest part of a partition. St000230Sum of the minimal elements of the blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000310The minimal degree of a vertex of a graph. 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. St000988The orbit size of a permutation under Foata's bijection. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001117The game chromatic index of a graph. St001118The acyclic chromatic index of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001341The number of edges in the center of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001468The smallest fixpoint of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001649The length of a longest trail in a graph. St001694The number of maximal dissociation sets in a graph. St001770The number of facets of a certain subword complex associated with the signed permutation. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001827The number of two-component spanning forests of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001933The largest multiplicity of a part in an integer partition. St000086The number of subgraphs. St000468The Hosoya index of a graph. St000822The Hadwiger number of the graph. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000570The Edelman-Greene number of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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 $eA$. St001545The second Elser number of a connected graph. 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. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000477The weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000871The number of very big ascents of a permutation. St001862The number of crossings of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000366The number of double descents of a permutation. St000497The lcb statistic of a set partition. St000562The number of internal points 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. 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. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000007The number of saliances of the permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000035The number of left outer peaks of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001524The degree of symmetry of a binary word. St001705The number of occurrences of the pattern 2413 in a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001413Half the length of the longest even length palindromic prefix of a binary word. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000662The staircase size of the code of a permutation. St000068The number of minimal elements in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St000741The Colin de Verdière graph invariant. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001058The breadth of the ordered tree. St001372The length of a longest cyclic run of ones of a binary word. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000406The number of occurrences of the pattern 3241 in a permutation. St000872The number of very big descents of a permutation. St000023The number of inner peaks of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000989The number of final rises of a permutation. St001130The number of two successive successions in a permutation. St001306The number of induced paths on four vertices in a graph. St001403The number of vertical separators in a permutation. St001434The number of negative sum pairs of a signed permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. 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. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000401The size of the symmetry class of a permutation. St001624The breadth of a lattice. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001845The number of join irreducibles minus the rank of a lattice. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000694The number of affine bounded permutations that project to a given permutation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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