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Your data matches 272 different statistics following compositions of up to 3 maps.
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Matching statistic: St001596
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St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> 0 = 1 - 1
[[2],[]]
=> 0 = 1 - 1
[[1,1],[]]
=> 0 = 1 - 1
[[2,1],[1]]
=> 0 = 1 - 1
[[3],[]]
=> 0 = 1 - 1
[[2,1],[]]
=> 0 = 1 - 1
[[3,1],[1]]
=> 0 = 1 - 1
[[2,2],[1]]
=> 0 = 1 - 1
[[3,2],[2]]
=> 0 = 1 - 1
[[1,1,1],[]]
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> 0 = 1 - 1
[[2,1,1],[1]]
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> 0 = 1 - 1
[[4],[]]
=> 0 = 1 - 1
[[3,1],[]]
=> 0 = 1 - 1
[[4,1],[1]]
=> 0 = 1 - 1
[[2,2],[]]
=> 1 = 2 - 1
[[3,2],[1]]
=> 0 = 1 - 1
[[4,2],[2]]
=> 0 = 1 - 1
[[2,1,1],[]]
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> 0 = 1 - 1
[[3,1,1],[1]]
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> 0 = 1 - 1
[[3,3],[2]]
=> 0 = 1 - 1
[[4,3],[3]]
=> 0 = 1 - 1
[[2,2,1],[1]]
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> 0 = 1 - 1
[[3,2,1],[2]]
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> 0 = 1 - 1
[[5],[]]
=> 0 = 1 - 1
[[4,1],[]]
=> 0 = 1 - 1
[[5,1],[1]]
=> 0 = 1 - 1
[[4,2],[1]]
=> 0 = 1 - 1
[[5,2],[2]]
=> 0 = 1 - 1
[[3,1,1],[]]
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> 0 = 1 - 1
[[4,1,1],[1]]
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> 0 = 1 - 1
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001633
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(load all 2 compositions to match this statistic)
Mp00185: Skew partitions —cell poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001476
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(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).
Matching statistic: St001305
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(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The number of induced cycles on four vertices in a graph.
Matching statistic: St001311
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001324
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph.
A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph.
A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge and $(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001331
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The size of the minimal feedback vertex set.
A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001335
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1 = 2 - 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0 = 1 - 1
[[5,2,1],[2,1]]
=> ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
The following 262 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001638The book thickness of a graph. St001736The total number of cycles in a graph. St000897The number of different multiplicities of parts of an integer partition. St001271The competition number of a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001871The number of triconnected components of a graph. St000379The number of Hamiltonian cycles in a graph. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000047The number of standard immaculate tableaux of a given shape. St000056The decomposition (or block) number of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000889The number of alternating sign matrices with the same antidiagonal sums. St001162The minimum jump of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001344The neighbouring number of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000042The number of crossings of a perfect matching. St000119The number of occurrences of the pattern 321 in a permutation. St000221The number of strong fixed points of a permutation. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000542The number of left-to-right-minima of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000750The number of occurrences of the pattern 4213 in a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000842The breadth of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000974The length of the trunk of an ordered tree. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001394The genus of a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001584The area statistic between a Dyck path and its bounce path. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001696The natural major index of a standard Young tableau. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000701The protection number of a binary tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001732The number of peaks visible from the left. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St000068The number of minimal elements in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001651The Frankl number of a lattice. St000115The single entry in the last row. St000764The number of strong records in an integer composition. St000769The major index of a composition regarded as a word. St000763The sum of the positions of the strong records of an integer composition. St000761The number of ascents in an integer composition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000914The sum of the values of the Möbius function of a poset. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001722The number of minimal chains with small intervals between a binary word and the top element. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000091The descent variation of a composition. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001890The maximum magnitude of the Möbius function of a poset. St001490The number of connected components of a skew partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001301The first Betti number of the order complex associated with the poset. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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