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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001636
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001636: Posets ⟶ 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] => ([(0,1)],2)
=> 2
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[2,4,1,3] => [3,4,1,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St001245
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[3,1,2] => [3,2,1] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,4,2,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => [4,1,3,2] => 3 = 4 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => [4,1,3,2] => 3 = 4 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [4,1,3,2] => 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => [4,3,1,2] => 2 = 3 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 3 = 4 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 2 = 3 - 1
[2,4,1,3] => [3,4,1,2] => [1,3,4,2] => [2,1,3,4] => 2 = 3 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 3 = 4 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 3 = 4 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 2 = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 2 = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => [1,2,4,3] => 2 = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4 = 5 - 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
Description
The cyclic maximal difference between two consecutive entries of a permutation.
This is given, for a permutation $\pi$ of length $n$, by
$$\max \{ |\pi(i) − \pi(i+1)| : 1 \leq i \leq n \}$$
where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000080
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 3 - 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[2,3,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 3 - 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[3,1,4,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[3,2,4,1] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
Description
The rank of the poset.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 1
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2
[2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,3,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,3,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[2,4,1,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,4,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[3,1,4,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[3,2,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[4,1,3,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[4,2,1,3] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,4,5,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,5,3,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,5,4,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,1,4,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,2,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,4,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,4,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,2,5,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,2,5,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,3,5,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,2,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,2,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,2,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,3,1,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,3,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,1,2,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,1,4,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,2,1,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,2,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,2,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,2,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2 - 1
[2,1] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[2,3,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,3,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,3,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,4,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[3,1,4,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[3,2,4,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,4,5,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,3,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,4,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,4,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,2,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,2,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,4,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,1,4] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,4,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001629
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Values
[1] => [[1]]
=> [1] => [1] => ? = 1 - 5
[1,2] => [[1,2]]
=> [2] => [1] => ? = 2 - 5
[2,1] => [[1],[2]]
=> [2] => [1] => ? = 2 - 5
[1,2,3] => [[1,2,3]]
=> [3] => [1] => ? = 3 - 5
[1,3,2] => [[1,2],[3]]
=> [3] => [1] => ? = 3 - 5
[2,1,3] => [[1,3],[2]]
=> [2,1] => [1,1] => ? = 3 - 5
[2,3,1] => [[1,3],[2]]
=> [2,1] => [1,1] => ? = 3 - 5
[3,1,2] => [[1,2],[3]]
=> [3] => [1] => ? = 3 - 5
[3,2,1] => [[1],[2],[3]]
=> [3] => [1] => ? = 3 - 5
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [1] => ? = 4 - 5
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => [1] => ? = 4 - 5
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 3 - 5
[1,3,4,2] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 4 - 5
[1,4,2,3] => [[1,2,3],[4]]
=> [4] => [1] => ? = 4 - 5
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 4 - 5
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 4 - 5
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => [2] => ? = 3 - 5
[2,3,1,4] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 4 - 5
[2,3,4,1] => [[1,3,4],[2]]
=> [2,2] => [2] => ? = 3 - 5
[2,4,1,3] => [[1,3],[2,4]]
=> [2,2] => [2] => ? = 3 - 5
[2,4,3,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 4 - 5
[3,1,2,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => ? = 4 - 5
[3,1,4,2] => [[1,2],[3,4]]
=> [3,1] => [1,1] => ? = 3 - 5
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 4 - 5
[3,2,4,1] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 3 - 5
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => [1,1] => ? = 4 - 5
[3,4,2,1] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => ? = 4 - 5
[4,1,2,3] => [[1,2,3],[4]]
=> [4] => [1] => ? = 3 - 5
[4,1,3,2] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 3 - 5
[4,2,1,3] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 4 - 5
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => ? = 4 - 5
[4,3,1,2] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 4 - 5
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [1] => ? = 4 - 5
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [1] => ? = 5 - 5
[2,4,5,3,1] => [[1,3,5],[2],[4]]
=> [2,2,1] => [2,1] => 0 = 5 - 5
[2,5,3,4,1] => [[1,3,4],[2],[5]]
=> [2,3] => [1,1] => ? = 5 - 5
[2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> [2,3] => [1,1] => ? = 5 - 5
[3,4,5,1,2] => [[1,2,5],[3,4]]
=> [3,2] => [1,1] => ? = 5 - 5
[3,4,5,2,1] => [[1,4,5],[2],[3]]
=> [3,2] => [1,1] => ? = 5 - 5
[3,5,1,4,2] => [[1,2],[3,4],[5]]
=> [3,2] => [1,1] => ? = 5 - 5
[3,5,2,4,1] => [[1,4],[2,5],[3]]
=> [3,2] => [1,1] => ? = 5 - 5
[3,5,4,1,2] => [[1,2],[3,4],[5]]
=> [3,2] => [1,1] => ? = 5 - 5
[3,5,4,2,1] => [[1,4],[2],[3],[5]]
=> [3,2] => [1,1] => ? = 5 - 5
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0 = 5 - 5
[4,2,5,3,1] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0 = 5 - 5
[4,3,5,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => [1,1] => ? = 5 - 5
[4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> [4,1] => [1,1] => ? = 5 - 5
[4,5,1,2,3] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => ? = 5 - 5
[4,5,1,3,2] => [[1,2],[3,5],[4]]
=> [4,1] => [1,1] => ? = 5 - 5
[4,5,2,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0 = 5 - 5
[4,5,2,3,1] => [[1,3],[2,5],[4]]
=> [2,2,1] => [2,1] => 0 = 5 - 5
[4,5,3,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => [1,1] => ? = 5 - 5
[4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> [4,1] => [1,1] => ? = 5 - 5
[5,2,3,1,4] => [[1,3,4],[2],[5]]
=> [2,3] => [1,1] => ? = 5 - 5
[5,2,3,4,1] => [[1,3,4],[2],[5]]
=> [2,3] => [1,1] => ? = 5 - 5
[2,4,6,5,3,1] => [[1,3,5],[2],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[2,5,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[2,5,6,4,3,1] => [[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[2,6,4,3,5,1] => [[1,3,5],[2],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[2,6,4,5,3,1] => [[1,3,5],[2],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[3,5,6,1,4,2] => [[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[3,5,6,2,4,1] => [[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[3,5,6,4,1,2] => [[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[3,5,6,4,2,1] => [[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[4,2,6,5,1,3] => [[1,3],[2,5],[4,6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,2,6,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,5,6,2,1,3] => [[1,3,6],[2,5],[4]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,2,1,5,3] => [[1,3],[2,5],[4,6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,2,3,5,1] => [[1,3,5],[2,6],[4]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,2,5,1,3] => [[1,3],[2,5],[4,6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,2,5,3,1] => [[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,5,2,1,3] => [[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[4,6,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> [2,2,2] => [3] => 1 = 6 - 5
[5,2,6,3,1,4] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,2,6,3,4,1] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,2,6,4,1,3] => [[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,2,6,4,3,1] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,1,2,4] => [[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,1,4,2] => [[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,2,1,4] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,2,4,1] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,4,1,2] => [[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,3,6,4,2,1] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,4,6,2,1,3] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,4,6,2,3,1] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,1,3,4] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,1,4,3] => [[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,3,1,4] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,4,1,3] => [[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,2,4,3,1] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,1,2,4] => [[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,1,4,2] => [[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,2,1,4] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,2,4,1] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,4,1,2] => [[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,3,4,2,1] => [[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 6 - 5
[5,6,4,2,1,3] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
[5,6,4,2,3,1] => [[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 6 - 5
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001637
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 3 - 1
[2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,3,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 3 - 1
[2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[3,2,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,4,3,5,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,4,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,5,3,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,5,4,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,4,6,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,4,6,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,1,4,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,2,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,4,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,4,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,1,5,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,2,5,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,1,4,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,2,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,4,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,4,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,2,6,5,1,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,2,6,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,3,6,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,3,6,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,1,2,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,1,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,2,1,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,2,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,3,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,6,2,1,5,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ? = 1 - 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[1,4,3,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 3 - 1
[2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[2,3,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[2,4,1,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 3 - 1
[2,4,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[3,1,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 4 - 1
[3,2,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[3,4,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,1,3,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[4,2,1,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,4,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,1,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,4,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,3,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,4,3,5,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,4,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,5,3,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[2,6,5,4,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,4,6,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,4,6,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,1,4,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,2,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,4,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,5,6,4,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,1,5,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,2,5,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,4,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,1,4,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,2,4,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,4,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[3,6,5,4,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,2,6,5,1,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,2,6,5,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,3,6,5,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,3,6,5,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,1,2,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,1,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,2,1,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,2,3,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,3,1,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,5,6,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[4,6,2,1,5,3] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001603
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1]
=> []
=> ? = 1 - 5
[1,2] => [1,2] => [2]
=> []
=> ? = 2 - 5
[2,1] => [1,2] => [2]
=> []
=> ? = 2 - 5
[1,2,3] => [1,2,3] => [3]
=> []
=> ? = 3 - 5
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 3 - 5
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 3 - 5
[2,3,1] => [1,2,3] => [3]
=> []
=> ? = 3 - 5
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 3 - 5
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? = 3 - 5
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 5
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 5
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 5
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 5
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 5
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 5
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 5
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 5
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 5
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 5
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 3 - 5
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 5
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 3 - 5
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 3 - 5
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 3 - 5
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 3 - 5
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 4 - 5
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 5
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 5
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 5
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 5 - 5
[2,4,5,3,1] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 5 - 5
[2,5,3,4,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 5 - 5
[2,5,4,3,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 5 - 5
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,4,1,2] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? = 5 - 5
[3,5,4,2,1] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? = 5 - 5
[4,2,5,1,3] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 5
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> ? = 5 - 5
[4,3,5,1,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 5
[4,3,5,2,1] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 5
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 5 - 5
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> ? = 5 - 5
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[4,6,2,3,5,1] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,2,6,3,4,1] => [1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,4,6,1,3,2] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,4,6,2,3,1] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,2,1,3,4] => [1,5,3,2,6,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,2,1,4,3] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,2,3,1,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,2,3,4,1] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[5,6,2,4,3,1] => [1,5,3,2,6,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,3,1,2,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,3,1,4,2] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,3,2,1,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[5,6,3,2,4,1] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,2,5,3,1,4] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,2,5,4,1,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,4,2,1,3,5] => [1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,4,2,3,1,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,4,3,1,2,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,4,3,2,1,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,4,5,1,2,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,4,5,2,1,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,2,1,3,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,2,1,4,3] => [1,6,3,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,2,3,1,4] => [1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,5,2,3,4,1] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,5,2,4,1,3] => [1,6,3,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,3,1,2,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,3,1,4,2] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[6,5,3,2,1,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 6 - 5
[6,5,3,2,4,1] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 18%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1]
=> []
=> ? = 1 - 6
[1,2] => [1,2] => [2]
=> []
=> ? = 2 - 6
[2,1] => [1,2] => [2]
=> []
=> ? = 2 - 6
[1,2,3] => [1,2,3] => [3]
=> []
=> ? = 3 - 6
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 3 - 6
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 3 - 6
[2,3,1] => [1,2,3] => [3]
=> []
=> ? = 3 - 6
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 3 - 6
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? = 3 - 6
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 6
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 6
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 6
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 6
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 6
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 6
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 6
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 6
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 4 - 6
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? = 3 - 6
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 3 - 6
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 4 - 6
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 6
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 3 - 6
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 6
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 3 - 6
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 6
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 4 - 6
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 3 - 6
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 3 - 6
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 4 - 6
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 6
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 6
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 4 - 6
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 5 - 6
[2,4,5,3,1] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 5 - 6
[2,5,3,4,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 5 - 6
[2,5,4,3,1] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 5 - 6
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 6
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 6
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 6
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 6
[3,5,4,1,2] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? = 5 - 6
[3,5,4,2,1] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? = 5 - 6
[4,2,5,1,3] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 6
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> ? = 5 - 6
[4,3,5,1,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 6
[4,3,5,2,1] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 5 - 6
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 5 - 6
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> ? = 5 - 6
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 5 - 6
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[4,6,2,3,5,1] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,2,6,3,4,1] => [1,5,4,3,6,2] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,4,6,1,3,2] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,4,6,2,3,1] => [1,5,3,6,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,2,1,3,4] => [1,5,3,2,6,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,2,1,4,3] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,2,3,1,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,2,3,4,1] => [1,5,4,3,2,6] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[5,6,2,4,3,1] => [1,5,3,2,6,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,3,1,2,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,3,1,4,2] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,3,2,1,4] => [1,5,2,6,4,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[5,6,3,2,4,1] => [1,5,4,2,6,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,2,5,3,1,4] => [1,6,4,3,5,2] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,2,5,4,1,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,4,2,1,3,5] => [1,6,5,3,2,4] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,4,2,3,1,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,4,3,1,2,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,4,3,2,1,5] => [1,6,5,2,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,4,5,1,2,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,4,5,2,1,3] => [1,6,3,5,2,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,2,1,3,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,2,1,4,3] => [1,6,3,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,2,3,1,4] => [1,6,4,3,2,5] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,5,2,3,4,1] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,5,2,4,1,3] => [1,6,3,2,5,4] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,3,1,2,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,3,1,4,2] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
[6,5,3,2,1,4] => [1,6,4,2,5,3] => [3,2,1]
=> [2,1]
=> 0 = 6 - 6
[6,5,3,2,4,1] => [1,6,2,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0 = 6 - 6
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
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