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Your data matches 138 different statistics following compositions of up to 3 maps.
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Matching statistic: St000217
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000218
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => 0
[2,1,3] => [1,3,2] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 0
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 1
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 0
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 1
[2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 0
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 0
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 0
[3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 0
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 2
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 0
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 0
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 0
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 0
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 0
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => 2
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St001398
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2,1] => ([],2)
=> 0
[2,1] => [1,2] => [2,1] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 0
[2,1,3] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 0
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 0
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 0
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 0
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 0
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 0
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St000219
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => ? ∊ {0,0}
[2,1] => [1,2] => [1,2] => ? ∊ {0,0}
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 3
[1,5,2,4,3] => [1,5,2,4,3] => [1,3,5,4,2] => 2
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St000567
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,5,3,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 0
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[4,3,2,5,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[4,3,5,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 0
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000714
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
[1] => [] => []
=> ?
=> ? = 0
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 3
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 3
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 3
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,1,5] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,4,5,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,3,5,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,3,5] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 3
[2,5,3,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 3
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 3
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 3
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 3
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
=> [2]
=> 3
[4,3,2,5,1] => [4,3,2,1] => [2,2]
=> [2]
=> 3
[4,3,5,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 3
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 3
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 3
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000771
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 57%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0} + 1
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0} + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
[3,1,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[3,4,2,1] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,1,5,3,2] => [1,4,3,5,2] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,2,5,3,1] => [1,4,3,5,2] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[4,5,3,2,1] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,1,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001964
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 57%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [2,1] => [2,1] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0}
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [2,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0}
[3,1,2] => [2,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0}
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[1,4,2,3] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[3,1,2,4] => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,4,2] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 1
[1,5,2,4,3] => [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,5,3,4,2] => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[2,1,4,5,3] => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,3,4] => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[2,3,1,5,4] => [2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[2,4,5,3,1] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,4,3] => [5,3,1,4,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,3,1] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,5,4] => [3,2,1,5,4] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [5,3,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [5,4,2,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,2,3] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,3,2] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,3,5,1,2] => [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,1,3,2] => [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,3,2,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,3,4,2] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,4,2,3] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,4,3,2] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,2,4,1,3] => [5,4,2,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,2,4,3,1] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,3,2,1,4] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,3,4,2,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,1,3,2] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,2,3,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,3,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,1,2,5,3,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,6,5,2,3,4] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,4,6,5,3] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => ([(1,5),(2,3),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [4,1,2,5,3,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000938
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [-1] => []
=> ? = 0
[1,2] => [1,2] => [2,-1] => []
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,-2] => [1]
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0,0,0}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> ? ∊ {0,0,0,0}
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> ? ∊ {0,0,0,0}
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0,0,0}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 0
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 0
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 0
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 0
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,1,1,1]
=> 0
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,1]
=> 0
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [2,1,1]
=> 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,1]
=> 0
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [2,1]
=> 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [2,1,1]
=> 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [3,1]
=> 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [2,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [3,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,1]
=> 0
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [2,1,1]
=> 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [1,1]
=> 0
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [2]
=> 0
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1]
=> 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1]
=> 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
The following 128 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000454The largest eigenvalue of a graph if it is integral. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000456The monochromatic index of a connected graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a 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. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001877Number of indecomposable injective modules with projective dimension 2. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001862The number of crossings of a signed permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. 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 $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001864The number of excedances of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000259The diameter 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001866The nesting alignments of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. 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 $K[x]/(x^n)$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 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. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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