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Your data matches 163 different statistics following compositions of up to 3 maps.
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Matching statistic: St000883
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => 2
[1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 2
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000363
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 1
Description
The number of minimal vertex covers of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices.
This is also the leading coefficient of the clique polynomial of the complement of $G$.
This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000909
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000911
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000911: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000911: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,2] => [2,1] => ([],2)
=> 1
[1,1,0,0]
=> [2,1] => [1,2] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => ([],3)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St001199
(load all 84 compositions to match this statistic)
(load all 84 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 80%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,2}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4}
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [6,5,2,4,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [6,3,5,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,1,4,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [6,5,1,2,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [6,3,5,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,1,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [6,1,5,2,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => [6,2,1,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [6,2,3,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [6,5,2,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,5,6,6,6,6,8}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001009
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001009: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 76%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001009: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 76%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 2
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 4
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 4
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,4,6,6,6,6,8}
Description
Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000208
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,5,6,6,6,8}
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000015
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
Description
The number of peaks of a Dyck path.
Matching statistic: St000288
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> => ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> => ? ∊ {1,2}
[1,1,0,0]
=> [[2],[]]
=> []
=> => ? ∊ {1,2}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> => ? ∊ {1,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> => ? ∊ {1,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> => ? ∊ {1,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> => ? ∊ {1,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> => ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1000 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 100 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 100 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> => ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1000 => 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000443
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 68%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,2,3,3,4}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,8}
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
The following 153 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001462The number of factors of a standard tableaux under concatenation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001933The largest multiplicity of a part in an integer partition. St001432The order dimension of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001481The minimal height of a peak of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000668The least common multiple of the parts of the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000327The number of cover relations in a poset. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000729The minimal arc length of a set partition. St001052The length of the exterior of a permutation. St000307The number of rowmotion orbits 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000910The number of maximal chains of minimal length in a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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 $eA$. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001118The acyclic chromatic index of a graph. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000145The Dyson rank of a partition. 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. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000454The largest eigenvalue of a graph if it is integral. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000635The number of strictly order preserving maps of a poset into itself. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St001879The 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. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000264The girth of a graph, which is not a tree. 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000663The number of right floats of a permutation. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice. St000648The number of 2-excedences of a permutation. St000366The number of double descents of a permutation.
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