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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001142
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Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001812
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(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 95%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,6,5,4] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,6,5,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,6,5,3] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,3,6,5,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,6,5,3,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,2,6,4,3] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,3,6,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,6,4,3,2] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,6,2,5,4,3] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,6,3,2,5,4] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,6,3,5,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,6,5,4,3,2] => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[4,3,1,6,5,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[4,3,2,6,5,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[4,3,6,5,2,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,3,1,6,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,3,2,6,4,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,3,6,4,2,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,1,6,3,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,4,2,6,3,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,4,3,6,2,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[5,4,6,3,2,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,3,1,5,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,3,2,1,5,4] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,3,2,5,4,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,3,5,4,2,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,1,5,3,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,2,1,5,3] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,2,5,3,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,3,1,5,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,3,2,5,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,3,5,2,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,4,5,3,2,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,1,4,3,2] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,2,1,4,3] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,5,2,4,3,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,3,1,4,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,5,3,2,4,1] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[6,5,3,4,2,1] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,4,3,2,1] => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Matching statistic: St000806
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 80% ●values known / values provided: 93%●distinct values known / distinct values provided: 80%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 80% ●values known / values provided: 93%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1] => ? = 0 + 2
[1,2] => [2] => [2] => [1] => ? = 0 + 2
[2,1] => [2] => [2] => [1] => ? = 0 + 2
[1,2,3] => [3] => [3] => [1] => ? = 0 + 2
[1,3,2] => [1,2] => [2,1] => [1,1] => 3 = 1 + 2
[2,1,3] => [3] => [3] => [1] => ? = 0 + 2
[2,3,1] => [3] => [3] => [1] => ? = 0 + 2
[3,1,2] => [3] => [3] => [1] => ? = 0 + 2
[3,2,1] => [2,1] => [1,2] => [1,1] => 3 = 1 + 2
[1,2,3,4] => [4] => [4] => [1] => ? = 0 + 2
[1,2,4,3] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[1,3,2,4] => [1,3] => [3,1] => [1,1] => 3 = 1 + 2
[1,3,4,2] => [1,3] => [3,1] => [1,1] => 3 = 1 + 2
[1,4,2,3] => [1,3] => [3,1] => [1,1] => 3 = 1 + 2
[1,4,3,2] => [1,2,1] => [2,1,1] => [1,2] => 4 = 2 + 2
[2,1,3,4] => [4] => [4] => [1] => ? = 0 + 2
[2,1,4,3] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[2,3,1,4] => [4] => [4] => [1] => ? = 0 + 2
[2,3,4,1] => [4] => [4] => [1] => ? = 0 + 2
[2,4,1,3] => [4] => [4] => [1] => ? = 0 + 2
[2,4,3,1] => [3,1] => [1,3] => [1,1] => 3 = 1 + 2
[3,1,2,4] => [4] => [4] => [1] => ? = 0 + 2
[3,1,4,2] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[3,2,1,4] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[3,2,4,1] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[3,4,1,2] => [4] => [4] => [1] => ? = 0 + 2
[3,4,2,1] => [3,1] => [1,3] => [1,1] => 3 = 1 + 2
[4,1,2,3] => [4] => [4] => [1] => ? = 0 + 2
[4,1,3,2] => [3,1] => [1,3] => [1,1] => 3 = 1 + 2
[4,2,1,3] => [2,2] => [2,2] => [2] => 3 = 1 + 2
[4,2,3,1] => [3,1] => [1,3] => [1,1] => 3 = 1 + 2
[4,3,1,2] => [1,3] => [3,1] => [1,1] => 3 = 1 + 2
[4,3,2,1] => [1,2,1] => [2,1,1] => [1,2] => 4 = 2 + 2
[1,2,3,4,5] => [5] => [5] => [1] => ? = 0 + 2
[1,2,3,5,4] => [3,2] => [2,3] => [1,1] => 3 = 1 + 2
[1,2,4,3,5] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[1,2,4,5,3] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[1,2,5,3,4] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[1,2,5,4,3] => [2,2,1] => [2,1,2] => [1,1,1] => 4 = 2 + 2
[1,3,2,4,5] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,3,2,5,4] => [1,2,2] => [2,2,1] => [2,1] => 4 = 2 + 2
[1,3,4,2,5] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,3,4,5,2] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,3,5,2,4] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,3,5,4,2] => [1,3,1] => [3,1,1] => [1,2] => 4 = 2 + 2
[1,4,2,3,5] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,4,2,5,3] => [1,2,2] => [2,2,1] => [2,1] => 4 = 2 + 2
[1,4,3,2,5] => [1,2,2] => [2,2,1] => [2,1] => 4 = 2 + 2
[1,4,3,5,2] => [1,2,2] => [2,2,1] => [2,1] => 4 = 2 + 2
[1,4,5,2,3] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,4,5,3,2] => [1,3,1] => [3,1,1] => [1,2] => 4 = 2 + 2
[1,5,2,3,4] => [1,4] => [4,1] => [1,1] => 3 = 1 + 2
[1,5,2,4,3] => [1,3,1] => [3,1,1] => [1,2] => 4 = 2 + 2
[1,5,3,2,4] => [1,2,2] => [2,2,1] => [2,1] => 4 = 2 + 2
[1,5,3,4,2] => [1,3,1] => [3,1,1] => [1,2] => 4 = 2 + 2
[1,5,4,2,3] => [1,1,3] => [1,3,1] => [1,1,1] => 4 = 2 + 2
[1,5,4,3,2] => [1,1,2,1] => [1,2,1,1] => [1,1,2] => 5 = 3 + 2
[2,1,3,4,5] => [5] => [5] => [1] => ? = 0 + 2
[2,1,3,5,4] => [3,2] => [2,3] => [1,1] => 3 = 1 + 2
[2,1,4,3,5] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[2,1,4,5,3] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[2,1,5,3,4] => [2,3] => [3,2] => [1,1] => 3 = 1 + 2
[2,1,5,4,3] => [2,2,1] => [2,1,2] => [1,1,1] => 4 = 2 + 2
[2,3,1,4,5] => [5] => [5] => [1] => ? = 0 + 2
[2,3,1,5,4] => [3,2] => [2,3] => [1,1] => 3 = 1 + 2
[2,3,4,1,5] => [5] => [5] => [1] => ? = 0 + 2
[2,3,4,5,1] => [5] => [5] => [1] => ? = 0 + 2
[2,3,5,1,4] => [5] => [5] => [1] => ? = 0 + 2
[2,3,5,4,1] => [4,1] => [1,4] => [1,1] => 3 = 1 + 2
[2,4,1,3,5] => [5] => [5] => [1] => ? = 0 + 2
[2,4,1,5,3] => [3,2] => [2,3] => [1,1] => 3 = 1 + 2
[2,4,3,1,5] => [3,2] => [2,3] => [1,1] => 3 = 1 + 2
[2,4,5,1,3] => [5] => [5] => [1] => ? = 0 + 2
[2,5,1,3,4] => [5] => [5] => [1] => ? = 0 + 2
[3,1,2,4,5] => [5] => [5] => [1] => ? = 0 + 2
[3,4,1,2,5] => [5] => [5] => [1] => ? = 0 + 2
[3,4,5,1,2] => [5] => [5] => [1] => ? = 0 + 2
[3,5,1,2,4] => [5] => [5] => [1] => ? = 0 + 2
[4,1,2,3,5] => [5] => [5] => [1] => ? = 0 + 2
[4,5,1,2,3] => [5] => [5] => [1] => ? = 0 + 2
[5,1,2,3,4] => [5] => [5] => [1] => ? = 0 + 2
[1,2,3,4,5,6] => [6] => [6] => [1] => ? = 0 + 2
[2,1,3,4,5,6] => [6] => [6] => [1] => ? = 0 + 2
[2,3,1,4,5,6] => [6] => [6] => [1] => ? = 0 + 2
[2,3,4,1,5,6] => [6] => [6] => [1] => ? = 0 + 2
[2,3,4,5,1,6] => [6] => [6] => [1] => ? = 0 + 2
[2,3,4,5,6,1] => [6] => [6] => [1] => ? = 0 + 2
[2,3,4,6,1,5] => [6] => [6] => [1] => ? = 0 + 2
[2,3,5,1,4,6] => [6] => [6] => [1] => ? = 0 + 2
[2,3,5,6,1,4] => [6] => [6] => [1] => ? = 0 + 2
[2,3,6,1,4,5] => [6] => [6] => [1] => ? = 0 + 2
[2,4,1,3,5,6] => [6] => [6] => [1] => ? = 0 + 2
[2,4,5,1,3,6] => [6] => [6] => [1] => ? = 0 + 2
[2,4,5,6,1,3] => [6] => [6] => [1] => ? = 0 + 2
[2,4,6,1,3,5] => [6] => [6] => [1] => ? = 0 + 2
[2,5,1,3,4,6] => [6] => [6] => [1] => ? = 0 + 2
[2,5,6,1,3,4] => [6] => [6] => [1] => ? = 0 + 2
[2,6,1,3,4,5] => [6] => [6] => [1] => ? = 0 + 2
[3,1,2,4,5,6] => [6] => [6] => [1] => ? = 0 + 2
[3,4,1,2,5,6] => [6] => [6] => [1] => ? = 0 + 2
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St001330
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,3,4] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[3,1,4,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,2,5] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,4,3,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,5,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,4,3,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,4,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,1,5,4] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,5,4,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,4,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,3,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,1,5,3] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,5,3,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,1,5,2] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,2,1,5] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,2,5,1] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,3,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,4,3,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,1,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,2,4,3,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,3,2,1,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,4,2,1,3] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,4,3,1,2] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,4,3,2,1] => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,2,3,6,5,4] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,4,3,6,5] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,4,6,5,3] => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,3,6,4] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,4,3,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,4,6,3] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,5,6,4,3] => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,3,5,4] => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,4,3,5] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,4,5,3] => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,5,3,4] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,6,5,4,3] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,3,2,4,6,5] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,4,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,2,6,5,4] => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,4,2,6,5] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,2,6,4] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,4,2,6] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,5,4,6,2] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,4,2,5] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,3,6,5,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001427
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 80%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001427: Signed permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [-1] => 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [-1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2,-1] => 1 = 0 + 1
[1,2,3] => [2,3,1] => [2,3,1] => [-1,2,3] => 1 = 0 + 1
[1,3,2] => [3,2,1] => [3,2,1] => [-1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [1,3,2] => [2,-1,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,-1] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,-1,2] => 1 = 0 + 1
[3,2,1] => [2,1,3] => [2,1,3] => [3,2,-1] => 2 = 1 + 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => [-1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => [-1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => [-1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [-1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => [-1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2 = 1 + 1
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 1 = 0 + 1
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 1 = 0 + 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => [3,-1,2,4] => 1 = 0 + 1
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => [3,-1,4,2] => 2 = 1 + 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [3,2,-1,4] => 2 = 1 + 1
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 2 = 1 + 1
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 1 = 0 + 1
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => [3,4,2,-1] => 2 = 1 + 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 1 = 0 + 1
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => [4,-1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => [4,2,3,-1] => 2 = 1 + 1
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => [4,3,-1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 3 = 2 + 1
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => [-1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => [-1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => [-1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => [-1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => [-1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => [-1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => [-1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => [-1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => [-1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [-1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [-1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [-1,3,5,4,2] => 3 = 2 + 1
[1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => [-1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [-1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => [-1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [-1,4,3,5,2] => 3 = 2 + 1
[1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => [-1,4,5,2,3] => 2 = 1 + 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [-1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,4,6,5] => [2,3,4,6,5,1] => [2,3,4,6,5,1] => [-1,2,3,4,6,5] => ? = 1 + 1
[1,2,3,5,4,6] => [2,3,5,4,6,1] => [2,3,5,4,6,1] => [-1,2,3,5,4,6] => ? = 1 + 1
[1,2,3,5,6,4] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => [-1,2,3,5,6,4] => ? = 1 + 1
[1,2,3,6,4,5] => [2,3,5,6,4,1] => [2,3,5,6,4,1] => [-1,2,3,6,4,5] => ? = 1 + 1
[1,2,3,6,5,4] => [2,3,6,5,4,1] => [2,3,6,5,4,1] => [-1,2,3,6,5,4] => ? = 2 + 1
[1,2,4,3,5,6] => [2,4,3,5,6,1] => [2,4,3,5,6,1] => [-1,2,4,3,5,6] => ? = 1 + 1
[1,2,4,3,6,5] => [2,4,3,6,5,1] => [2,4,3,6,5,1] => [-1,2,4,3,6,5] => ? = 2 + 1
[1,2,4,5,3,6] => [2,5,3,4,6,1] => [2,5,3,4,6,1] => [-1,2,4,5,3,6] => ? = 1 + 1
[1,2,4,5,6,3] => [2,6,3,4,5,1] => [2,6,3,4,5,1] => [-1,2,4,5,6,3] => ? = 1 + 1
[1,2,4,6,3,5] => [2,5,3,6,4,1] => [2,5,3,6,4,1] => [-1,2,4,6,3,5] => ? = 1 + 1
[1,2,4,6,5,3] => [2,6,3,5,4,1] => [2,6,3,5,4,1] => [-1,2,4,6,5,3] => ? = 2 + 1
[1,2,5,3,4,6] => [2,4,5,3,6,1] => [2,4,5,3,6,1] => [-1,2,5,3,4,6] => ? = 1 + 1
[1,2,5,3,6,4] => [2,4,6,3,5,1] => [2,4,6,3,5,1] => [-1,2,5,3,6,4] => ? = 2 + 1
[1,2,5,4,3,6] => [2,5,4,3,6,1] => [2,5,4,3,6,1] => [-1,2,5,4,3,6] => ? = 2 + 1
[1,2,5,4,6,3] => [2,6,4,3,5,1] => [2,6,4,3,5,1] => [-1,2,5,4,6,3] => ? = 2 + 1
[1,2,5,6,3,4] => [2,5,6,3,4,1] => [2,5,6,3,4,1] => [-1,2,5,6,3,4] => ? = 1 + 1
[1,2,5,6,4,3] => [2,6,5,3,4,1] => [2,6,5,3,4,1] => [-1,2,5,6,4,3] => ? = 2 + 1
[1,2,6,3,4,5] => [2,4,5,6,3,1] => [2,4,5,6,3,1] => [-1,2,6,3,4,5] => ? = 1 + 1
[1,2,6,3,5,4] => [2,4,6,5,3,1] => [2,4,6,5,3,1] => [-1,2,6,3,5,4] => ? = 2 + 1
[1,2,6,4,3,5] => [2,5,4,6,3,1] => [2,5,4,6,3,1] => [-1,2,6,4,3,5] => ? = 2 + 1
[1,2,6,4,5,3] => [2,6,4,5,3,1] => [2,6,4,5,3,1] => [-1,2,6,4,5,3] => ? = 2 + 1
[1,2,6,5,3,4] => [2,5,6,4,3,1] => [2,5,6,4,3,1] => [-1,2,6,5,3,4] => ? = 2 + 1
[1,2,6,5,4,3] => [2,6,5,4,3,1] => [2,6,5,4,3,1] => [-1,2,6,5,4,3] => ? = 3 + 1
[1,3,2,4,5,6] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => [-1,3,2,4,5,6] => ? = 1 + 1
[1,3,2,4,6,5] => [3,2,4,6,5,1] => [3,2,4,6,5,1] => [-1,3,2,4,6,5] => ? = 2 + 1
[1,3,2,5,4,6] => [3,2,5,4,6,1] => [3,2,5,4,6,1] => [-1,3,2,5,4,6] => ? = 2 + 1
[1,3,2,5,6,4] => [3,2,6,4,5,1] => [3,2,6,4,5,1] => [-1,3,2,5,6,4] => ? = 2 + 1
[1,3,2,6,4,5] => [3,2,5,6,4,1] => [3,2,5,6,4,1] => [-1,3,2,6,4,5] => ? = 2 + 1
[1,3,2,6,5,4] => [3,2,6,5,4,1] => [3,2,6,5,4,1] => [-1,3,2,6,5,4] => ? = 2 + 1
[1,3,4,2,5,6] => [4,2,3,5,6,1] => [4,2,3,5,6,1] => [-1,3,4,2,5,6] => ? = 1 + 1
[1,3,4,2,6,5] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => [-1,3,4,2,6,5] => ? = 2 + 1
[1,3,4,5,2,6] => [5,2,3,4,6,1] => [5,2,3,4,6,1] => [-1,3,4,5,2,6] => ? = 1 + 1
[1,3,4,5,6,2] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => [-1,3,4,5,6,2] => ? = 1 + 1
[1,3,4,6,2,5] => [5,2,3,6,4,1] => [5,2,3,6,4,1] => [-1,3,4,6,2,5] => ? = 1 + 1
[1,3,4,6,5,2] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => [-1,3,4,6,5,2] => ? = 2 + 1
[1,3,5,2,4,6] => [4,2,5,3,6,1] => [4,2,5,3,6,1] => [-1,3,5,2,4,6] => ? = 1 + 1
[1,3,5,2,6,4] => [4,2,6,3,5,1] => [4,2,6,3,5,1] => [-1,3,5,2,6,4] => ? = 2 + 1
[1,3,5,4,2,6] => [5,2,4,3,6,1] => [5,2,4,3,6,1] => [-1,3,5,4,2,6] => ? = 2 + 1
[1,3,5,4,6,2] => [6,2,4,3,5,1] => [6,2,4,3,5,1] => [-1,3,5,4,6,2] => ? = 2 + 1
[1,3,5,6,2,4] => [5,2,6,3,4,1] => [5,2,6,3,4,1] => [-1,3,5,6,2,4] => ? = 1 + 1
[1,3,5,6,4,2] => [6,2,5,3,4,1] => [6,2,5,3,4,1] => [-1,3,5,6,4,2] => ? = 2 + 1
[1,3,6,2,4,5] => [4,2,5,6,3,1] => [4,2,5,6,3,1] => [-1,3,6,2,4,5] => ? = 1 + 1
[1,3,6,2,5,4] => [4,2,6,5,3,1] => [4,2,6,5,3,1] => [-1,3,6,2,5,4] => ? = 2 + 1
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [5,2,4,6,3,1] => [-1,3,6,4,2,5] => ? = 2 + 1
[1,3,6,4,5,2] => [6,2,4,5,3,1] => [6,2,4,5,3,1] => [-1,3,6,4,5,2] => ? = 2 + 1
[1,3,6,5,2,4] => [5,2,6,4,3,1] => [5,2,6,4,3,1] => [-1,3,6,5,2,4] => ? = 2 + 1
[1,3,6,5,4,2] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => [-1,3,6,5,4,2] => ? = 2 + 1
[1,4,2,3,5,6] => [3,4,2,5,6,1] => [3,4,2,5,6,1] => [-1,4,2,3,5,6] => ? = 1 + 1
[1,4,2,3,6,5] => [3,4,2,6,5,1] => [3,4,2,6,5,1] => [-1,4,2,3,6,5] => ? = 2 + 1
Description
The number of descents of a signed permutation.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
Matching statistic: St001960
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
St001960: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Values
[1] => ? = 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 2
[1,2,3,4,5,6] => ? = 0
[1,2,3,4,6,5] => ? = 1
[1,2,3,5,4,6] => ? = 1
[1,2,3,5,6,4] => ? = 1
[1,2,3,6,4,5] => ? = 1
[1,2,3,6,5,4] => ? = 2
[1,2,4,3,5,6] => ? = 1
[1,2,4,3,6,5] => ? = 2
[1,2,4,5,3,6] => ? = 1
[1,2,4,5,6,3] => ? = 1
[1,2,4,6,3,5] => ? = 1
[1,2,4,6,5,3] => ? = 2
[1,2,5,3,4,6] => ? = 1
[1,2,5,3,6,4] => ? = 2
[1,2,5,4,3,6] => ? = 2
[1,2,5,4,6,3] => ? = 2
[1,2,5,6,3,4] => ? = 1
[1,2,5,6,4,3] => ? = 2
[1,2,6,3,4,5] => ? = 1
[1,2,6,3,5,4] => ? = 2
[1,2,6,4,3,5] => ? = 2
[1,2,6,4,5,3] => ? = 2
[1,2,6,5,3,4] => ? = 2
[1,2,6,5,4,3] => ? = 3
[1,3,2,4,5,6] => ? = 1
[1,3,2,4,6,5] => ? = 2
[1,3,2,5,4,6] => ? = 2
[1,3,2,5,6,4] => ? = 2
[1,3,2,6,4,5] => ? = 2
[1,3,2,6,5,4] => ? = 2
[1,3,4,2,5,6] => ? = 1
[1,3,4,2,6,5] => ? = 2
[1,3,4,5,2,6] => ? = 1
[1,3,4,5,6,2] => ? = 1
[1,3,4,6,2,5] => ? = 1
[1,3,4,6,5,2] => ? = 2
[1,3,5,2,4,6] => ? = 1
[1,3,5,2,6,4] => ? = 2
[1,3,5,4,2,6] => ? = 2
[1,3,5,4,6,2] => ? = 2
[1,3,5,6,2,4] => ? = 1
[1,3,5,6,4,2] => ? = 2
[1,3,6,2,4,5] => ? = 1
[1,3,6,2,5,4] => ? = 2
[1,3,6,4,2,5] => ? = 2
[1,3,6,4,5,2] => ? = 2
[1,3,6,5,2,4] => ? = 2
[1,3,6,5,4,2] => ? = 2
[1,4,2,3,5,6] => ? = 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001964
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 80%
Values
[1] => [.,.]
=> [[]]
=> ([(0,1)],2)
=> 0
[1,2] => [.,[.,.]]
=> [[[]]]
=> ([(0,2),(2,1)],3)
=> 0
[2,1] => [[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 2
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 2
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 2
[4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 2
[5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ? = 2
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[[]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> [[[[[[],[]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[[[]],[]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[[[[],[[]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> [[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ? = 2
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
=> [[[[[[]]],[]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 1
[1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
=> [[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[1,2,5,3,4,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,3,6,4] => [.,[.,[[.,.],[[.,.],.]]]]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,5,4,6,3] => [.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 1
[1,2,5,6,4,3] => [.,[.,[[[.,[.,.]],.],.]]]
=> [[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,6,3,4,5] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 1
[1,2,6,3,5,4] => [.,[.,[[.,.],[[.,.],.]]]]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 2
[1,2,6,4,3,5] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,6,4,5,3] => [.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,6,5,3,4] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 2
[1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> [[[[],[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ? = 3
[1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> [[[],[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> [[[],[[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ? = 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[1,3,2,5,6,4] => [.,[[.,.],[[.,[.,.]],.]]]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
=> [[[],[[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)
=> ? = 2
[1,3,4,2,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,4,2,6,5] => [.,[[.,[.,.]],[[.,.],.]]]
=> [[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
=> [[[[[[]]]],[]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 1
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
=> [[[[[],[]]],[]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ? = 2
[1,3,5,2,4,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,5,2,6,4] => [.,[[.,[.,.]],[[.,.],.]]]
=> [[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[[[],[]],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
[1,3,5,4,6,2] => [.,[[.,[[.,.],[.,.]]],.]]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[[[[]]],[[]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,5,6,4,2] => [.,[[.,[[.,[.,.]],.]],.]]
=> [[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2
[1,3,6,2,4,5] => [.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[]],[[[]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 1
[1,3,6,2,5,4] => [.,[[.,[.,.]],[[.,.],.]]]
=> [[[[]],[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
[1,3,6,4,2,5] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[[[],[]],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 2
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000307
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [3,1,2,4] => [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)
=> ? = 1 + 1
[1,3,4,2] => [3,1,4,2] => [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)
=> ? = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,4,2] => [3,4,1,2] => [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)
=> ? = 1 + 1
[3,2,1,4] => [3,2,1,4] => [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)
=> ? = 1 + 1
[3,2,4,1] => [3,2,4,1] => [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)
=> ? = 1 + 1
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [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)
=> ? = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 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)
=> 1 = 0 + 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,2,4,5,3] => [4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,2,5,4] => [5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,3,4,2,5] => [3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[1,3,4,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,2,4] => [3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [4,5,1,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,3,5,2] => [4,3,1,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,5,3,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [5,2,1,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,4,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,4,5,3] => [4,2,1,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,5,3,4] => [2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,1,5,4,3] => [5,4,2,1,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [5,2,3,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,3,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,3,5,1] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 + 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,4,1,3] => [2,5,1,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,4,3,1] => [5,4,2,3,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,1,2,5,4] => [5,1,3,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [3,1,2,4] => [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)
=> ? = 1 + 1
[1,3,4,2] => [3,1,4,2] => [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)
=> ? = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,4,2] => [3,4,1,2] => [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)
=> ? = 1 + 1
[3,2,1,4] => [3,2,1,4] => [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)
=> ? = 1 + 1
[3,2,4,1] => [3,2,4,1] => [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)
=> ? = 1 + 1
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [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)
=> ? = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 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)
=> 1 = 0 + 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,2,4,5,3] => [4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,2,5,4] => [5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,3,4,2,5] => [3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[1,3,4,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,2,4] => [3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,2,5,3] => [4,5,1,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,4,3,5,2] => [4,3,1,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,5,3,2,4] => [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [5,2,1,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,4,3,5] => [4,2,1,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,4,5,3] => [4,2,1,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,1,5,3,4] => [2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,1,5,4,3] => [5,4,2,1,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [5,2,3,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,1,5,3] => [4,2,5,1,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,3,1,5] => [4,2,3,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 + 1
[2,4,3,5,1] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 + 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,4,1,3] => [2,5,1,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,5,4,3,1] => [5,4,2,3,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001896
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [-1] => [-1] => 1 = 0 + 1
[1,2] => [1,2] => [2,-1] => [-2,-1] => 1 = 0 + 1
[2,1] => [2,1] => [1,-2] => [1,-2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,-1] => [3,-2,-1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,2,-1] => [-2,-3,-1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,3,-2] => [3,1,-2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,-3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,-2] => [1,-3,-2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1,-3] => [-2,1,-3] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [3,4,-2,-1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [3,-2,-4,-1] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [-2,4,-3,-1] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [2,3,-4,-1] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,-4,-3,-1] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [-2,-3,-4,-1] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [3,4,1,-2] => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [3,1,-4,-2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,4,2,-3] => 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,-4] => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,2,-4,-3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,-3,2,-4] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [1,4,-3,-2] => 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [-4,3,1,-2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [-2,4,1,-3] => 2 = 1 + 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [-2,1,3,-4] => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [-4,1,2,-3] => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,-4] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [-4,1,-3,-2] => 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [1,-3,-4,-2] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [-4,-2,1,-3] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3,-2,1,-4] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [-2,1,-4,-3] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [-2,-3,1,-4] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [3,4,5,-2,-1] => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [3,4,-2,-5,-1] => ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [3,-2,5,-4,-1] => ? = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [-2,3,4,-5,-1] => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [5,3,-2,-4,-1] => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [3,-2,-4,-5,-1] => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [-2,4,5,-3,-1] => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [-2,4,-3,-5,-1] => ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [2,3,5,-4,-1] => ? = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [5,3,4,-2,-1] => ? = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [-2,3,-5,-4,-1] => ? = 1 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,3,-4,-5,-1] => ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,-4,5,-3,-1] => ? = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [-5,2,4,-3,-1] => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [-2,-3,5,-4,-1] => ? = 2 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [-2,-5,4,-3,-1] => ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [5,2,3,-4,-1] => ? = 1 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [-4,2,3,-5,-1] => ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [5,2,-4,-3,-1] => ? = 1 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => [2,-4,-3,-5,-1] => ? = 2 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => [-5,2,-3,-4,-1] => ? = 2 + 1
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [4,2,-3,-5,-1] => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [2,-3,-5,-4,-1] => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [-2,-3,-4,-5,-1] => ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [3,4,5,1,-2] => ? = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [3,4,1,-5,-2] => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [3,1,5,-4,-2] => ? = 1 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,3,4,-5,-2] => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [5,3,1,-4,-2] => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [3,1,-4,-5,-2] => ? = 2 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,4,5,2,-3] => 1 = 0 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,4,2,-5,-3] => 2 = 1 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,5,3,-4] => 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,2,3,4,-5] => 1 = 0 + 1
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,2,3,-5,-4] => 1 = 0 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,-4,3,-5] => 2 = 1 + 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,2,5,-4,-3] => 1 = 0 + 1
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,-5,4,2,-3] => 2 = 1 + 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,-3,5,2,-4] => 2 = 1 + 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,-3,2,4,-5] => 2 = 1 + 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,-5,2,3,-4] => 1 = 0 + 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [-4,1,2,3,-5] => ? = 1 + 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [5,1,2,-4,-3] => ? = 0 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1,2,-4,-5,-3] => 2 = 1 + 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [5,1,-3,2,-4] => ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [1,4,-3,2,-5] => 2 = 1 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,-3,2,-5,-4] => 2 = 1 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [1,-3,-4,2,-5] => 3 = 2 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => [1,4,5,-3,-2] => 1 = 0 + 1
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [-4,3,5,1,-2] => ? = 1 + 1
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [5,3,4,1,-2] => ? = 1 + 1
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [-4,3,1,-5,-2] => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [-2,4,5,1,-3] => ? = 1 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [-2,4,1,-5,-3] => ? = 2 + 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [-2,1,5,3,-4] => ? = 1 + 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [-2,1,3,4,-5] => ? = 1 + 1
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => [-2,1,3,-5,-4] => ? = 1 + 1
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => [-2,1,-4,3,-5] => ? = 2 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => [-4,1,5,2,-3] => ? = 0 + 1
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [5,1,4,2,-3] => ? = 1 + 1
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [3,1,5,2,-4] => ? = 1 + 1
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [3,1,2,4,-5] => ? = 1 + 1
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [5,1,2,3,-4] => ? = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => [-4,1,2,-5,-3] => ? = 1 + 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => [-4,1,-3,2,-5] => ? = 1 + 1
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [3,1,2,-5,-4] => ? = 1 + 1
[3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,-5] => [3,1,-4,2,-5] => ? = 2 + 1
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
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