searching the database
Your data matches 641 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001496
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St001496: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001496: 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)
=> 1
Description
The number of graphs with the same Laplacian spectrum as the given graph.
Matching statistic: St000816
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2] => [1] => 1
[2,1] => [2] => [1] => 1
[1,2,3] => [3] => [1] => 1
[1,3,2] => [1,2] => [1,1] => 1
[2,1,3] => [3] => [1] => 1
[2,3,1] => [3] => [1] => 1
[3,1,2] => [3] => [1] => 1
[3,2,1] => [2,1] => [1,1] => 1
[1,2,3,4] => [4] => [1] => 1
[1,2,4,3] => [2,2] => [2] => 1
[1,3,2,4] => [1,3] => [1,1] => 1
[1,3,4,2] => [1,3] => [1,1] => 1
[1,4,2,3] => [1,3] => [1,1] => 1
[1,4,3,2] => [1,2,1] => [1,1,1] => 1
[2,1,3,4] => [4] => [1] => 1
[2,1,4,3] => [2,2] => [2] => 1
[2,3,1,4] => [4] => [1] => 1
[2,3,4,1] => [4] => [1] => 1
[2,4,1,3] => [4] => [1] => 1
[2,4,3,1] => [3,1] => [1,1] => 1
[3,1,2,4] => [4] => [1] => 1
[3,1,4,2] => [2,2] => [2] => 1
[3,2,1,4] => [2,2] => [2] => 1
[3,2,4,1] => [2,2] => [2] => 1
[3,4,1,2] => [4] => [1] => 1
[3,4,2,1] => [3,1] => [1,1] => 1
[4,1,2,3] => [4] => [1] => 1
[4,1,3,2] => [3,1] => [1,1] => 1
[4,2,1,3] => [2,2] => [2] => 1
[4,2,3,1] => [3,1] => [1,1] => 1
[4,3,1,2] => [1,3] => [1,1] => 1
[4,3,2,1] => [1,2,1] => [1,1,1] => 1
[1,2,3,4,5] => [5] => [1] => 1
[1,2,3,5,4] => [3,2] => [1,1] => 1
[1,2,4,3,5] => [2,3] => [1,1] => 1
[1,2,4,5,3] => [2,3] => [1,1] => 1
[1,2,5,3,4] => [2,3] => [1,1] => 1
[1,2,5,4,3] => [2,2,1] => [2,1] => 1
[1,3,2,4,5] => [1,4] => [1,1] => 1
[1,3,2,5,4] => [1,2,2] => [1,2] => 1
[1,3,4,2,5] => [1,4] => [1,1] => 1
[1,3,4,5,2] => [1,4] => [1,1] => 1
[1,3,5,2,4] => [1,4] => [1,1] => 1
[1,3,5,4,2] => [1,3,1] => [1,1,1] => 1
[1,4,2,3,5] => [1,4] => [1,1] => 1
[1,4,2,5,3] => [1,2,2] => [1,2] => 1
[1,4,3,2,5] => [1,2,2] => [1,2] => 1
[1,4,3,5,2] => [1,2,2] => [1,2] => 1
[1,4,5,2,3] => [1,4] => [1,1] => 1
Description
The number of standard composition tableaux of the composition.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
Matching statistic: St001518
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,2] => ([],2)
=> ([],1)
=> 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> ([],1)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => ([],5)
=> ([],1)
=> 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1
Description
The number of graphs with the same ordinary spectrum as the given graph.
Matching statistic: St000375
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00329: Permutations —Tanimoto⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,3,2] => [2,1,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[1,4,3,2] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 0 = 1 - 1
[2,1,4,3] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 0 = 1 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[3,2,1,4] => [1,4,2,3] => [2,1,3,4] => 0 = 1 - 1
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => 0 = 1 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,4,5] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,4,5,3] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,4,5,1,3] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,3,5] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,3,5] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,5,3,4] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [2,5,3,1,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,5,3,1,4] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,5,3,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,3,4] => 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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,1,3] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => 0 = 1 - 1
[2,1,4,3] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[2,3,1,4] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[3,2,1,4] => [1,4,2,3] => [4,1,2,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,5,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,2,5,4] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,5,2,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [4,1,2,5,3] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [4,1,2,5,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1,2,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => 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: St000440
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000440: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[2,1,3] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => 0 = 1 - 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => 0 = 1 - 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 0 = 1 - 1
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => 0 = 1 - 1
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => 0 = 1 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => 0 = 1 - 1
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => 0 = 1 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => 0 = 1 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => 0 = 1 - 1
Description
The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation.
There is a bijection between permutations avoiding these two pattern and Schröder paths [1,2].
Matching statistic: St000664
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [3,2,1,4] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [3,2,1,4] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [3,1,4,2] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [3,2,1,4] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [3,2,1,4] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [4,3,2,1] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [4,3,2,1] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [4,3,2,1] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [4,3,2,1] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [2,4,3,1] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,3,1,4] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [2,4,3,1] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,3,1,4] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [2,3,1,4] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [2,3,1,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [3,1,4,2,5] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [3,1,4,2,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [3,1,4,5,2] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [3,1,4,2,5] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [3,1,4,2,5] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,1,5,4,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,1,5,2,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,1,5,4,2] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,1,5,2,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,1,5,4,2] => 0 = 1 - 1
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000666
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [1,2,4,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
Description
The number of right tethers of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right tether is a large ascent between two consecutive rafts of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001550
(load all 51 compositions to match this statistic)
(load all 51 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 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: St000036
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,3,2] => [3,1,2] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,2,3] => 1
[2,1,3] => [2,1,3] => [2,3,1] => [1,2,3] => 1
[2,3,1] => [2,3,1] => [2,1,3] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[1,2,4,3] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[1,3,2,4] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [1,2,3,4] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => [1,3,2,4] => 1
[2,3,4,1] => [2,4,3,1] => [4,2,1,3] => [1,3,2,4] => 1
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [2,4,3,1] => [4,2,1,3] => [1,3,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => [1,4,2,3] => 1
[3,1,4,2] => [3,1,4,2] => [3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [3,2,1,4] => [3,4,2,1] => [1,2,3,4] => 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [1,2,3,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,2,1,4] => [1,4,2,3] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,3,1] => [1,2,4,3] => 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [1,2,3,4] => 1
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => [1,4,2,3] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => [1,2,3,4,5] => 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.
The following 631 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000805The number of peaks of the associated bargraph. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St001282The number of graphs with the same chromatic polynomial. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001740The number of graphs with the same symmetric edge polytope as the given graph. 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. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000233The number of nestings of a set partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000366The number of double descents of a permutation. St000370The genus of a graph. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000731The number of double exceedences of a permutation. St000768The number of peaks in an integer composition. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. 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. 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. 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. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001871The number of triconnected components of a graph. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000570The Edelman-Greene number of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000516The number of stretching pairs of a permutation. 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. 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. St000842The breadth of a permutation. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000449The number of pairs of vertices of a graph with distance 4. St000785The number of distinct colouring schemes of a graph. St000068The number of minimal elements in a poset. St000788The number of nesting-similar perfect matchings of a perfect matching. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St000787The number of flips required to make a perfect matching noncrossing. St000879The number of long braid edges in the graph of braid moves of a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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 $eA$. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000065The number of entries equal to -1 in an alternating sign matrix. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000447The number of pairs of vertices of a graph with distance 3. St000003The number of standard Young tableaux of the partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. 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. St000889The number of alternating sign matrices with the same antidiagonal sums. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St000481The number of upper covers of a partition in dominance order. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001272The number of graphs with the same degree sequence. 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. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest 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. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of 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. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex 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. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000069The number of maximal elements of a poset. St000914The sum of the values of the Möbius function of a poset. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000535The rank-width of a graph. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000352The Elizalde-Pak rank of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000546The number of global descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001626The number of maximal proper sublattices of a lattice. St001651The Frankl number of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St000234The number of global ascents of a permutation. St000284The Plancherel distribution on integer partitions. 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. 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. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. 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. 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. St001783The number of odd automorphisms of a graph. St000181The number of connected components of the Hasse diagram for the poset. 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. St000031The number of cycles in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. 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. 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 $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama 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. St001552The number of inversions between excedances and fixed points of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. 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 $Af$. St001964The interval resolution global dimension of a poset. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. St001271The competition number of a graph. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000210Minimum over maximum difference of elements in cycles. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001728The number of invisible descents of a permutation. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000143The largest repeated part of a partition. St000185The weighted size of a partition. St001214The aft of an integer partition. St000058The order of a permutation. St001442The number of standard Young tableaux whose major index is divisible by the size of a given 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. St000667The greatest common divisor of the parts of the partition. 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St001260The permanent of an alternating sign matrix. St001975The corank of the alternating sign matrix. St001011Number of simple modules of projective dimension 2 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. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000962The 3-shifted major index of a permutation. 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. St001025Number of simple modules with projective dimension 4 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. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001537The number of cyclic crossings of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000478Another weight of a partition according to Alladi. St000617The number of global maxima of a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. 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. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000935The number of ordered refinements of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St000878The number of ones minus the number of zeros of a binary word. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000629The defect of a binary word. St001095The number of non-isomorphic posets with precisely one further covering relation. St000010The length of the partition. St000048The multinomial of the parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000627The exponent of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000783The side length of the largest staircase partition fitting into a partition. St001372The length of a longest cyclic run of ones of a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001484The number of singletons of an integer partition. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000257The number of distinct parts of a partition that occur at least twice. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. 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. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct 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. St001961The sum of the greatest common divisors of all pairs of parts. St000007The number of saliances of the permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. 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. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000237The number of small exceedances. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of 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. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. 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. St000054The first entry of the permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000754The Grundy value for the game of removing nestings in a perfect matching. St001545The second Elser number of a connected graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000359The number of occurrences of the pattern 23-1. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000759The smallest missing part in an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001394The genus of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000717The number of ordinal summands of a poset. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000297The number of leading ones in a binary word. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001429The number of negative entries in a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000286The number of connected components of the complement of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St001430The number of positive entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000654The first descent of a permutation. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000843The decomposition number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000958The number of Bruhat factorizations of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000991The number of right-to-left minima of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001828The Euler characteristic of a graph. St000002The number of occurrences of the pattern 123 in a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000485The length of the longest cycle of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000549The number of odd partial sums of an integer partition. St000733The row containing the largest entry of a standard tableau. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000989The number of final rises of a permutation. St000995The largest even part of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001082The number of boxed occurrences of 123 in a permutation. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001252Half the sum of the even parts of a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001555The order of a signed permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances 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. St001248Sum of the even parts of a partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001621The number of atoms of a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000256The number of parts from which one can substract 2 and still get an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001624The breadth of a lattice. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001328The minimal number of occurrences of the bipartite-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. 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. St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001768The number of reduced words of a signed permutation. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. 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. 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. St000508Eigenvalues of the random-to-random operator acting on a simple module. 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. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001335The cardinality of a minimal cycle-isolating set of a graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001524The degree of symmetry of a binary word. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. 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. 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. St000264The girth of a graph, which is not a tree. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000782The indicator function of whether a given perfect matching is an L & P matching. St001570The minimal number of edges to add to make a graph Hamiltonian.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!