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Your data matches 405 different statistics following compositions of up to 3 maps.
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Matching statistic: St001596
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
[[3,2],[]]
=> 1 = 2 - 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
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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1 = 2 - 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
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001476
(load all 2 compositions to match this statistic)
(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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[[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
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).
Matching statistic: St001305
(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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 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
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 395 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. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000666The number of right tethers of a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001871The number of triconnected components of a graph. St000667The greatest common divisor of the parts of the partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001571The Cartan determinant of the integer partition. St001271The competition number of a graph. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000781The number of proper colouring schemes of a Ferrers diagram. St000655The length of the minimal rise of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. 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. St000183The side length of the Durfee square of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000913The number of ways to refine the partition into singletons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. 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. St000628The balance of a binary word. St000657The smallest part of an integer composition. St000659The number of rises of length at least 2 of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000711The number of big exceedences of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000816The number of standard composition tableaux of the composition. 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. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000905The number of different multiplicities of parts of an integer composition. St000920The logarithmic height of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001075The minimal size of a block of a set partition. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $K[x]/(x^n)$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001267The length of the Lyndon factorization of the binary word. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001590The crossing number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001665The number of pure excedances of a permutation. St001732The number of peaks visible from the left. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000899The maximal number of repetitions of an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000914The sum of the values of the Möbius function of a poset. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001597The Frobenius rank of a skew partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001394The genus of a permutation. St000842The breadth of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001592The maximal number of simple paths between any two different vertices of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001490The number of connected components of a skew partition. St000407The number of occurrences of the pattern 2143 in a permutation. St000768The number of peaks in an integer composition. 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. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. 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. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001513The number of nested exceedences of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. 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. St000042The number of crossings of a perfect matching. St000220The number of occurrences of the pattern 132 in a permutation. St000296The length of the symmetric border of a binary word. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000733The row containing the largest entry of a standard tableau. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in 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. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000331The number of upper interactions of a Dyck path. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000675The number of centered multitunnels of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001595The number of standard Young tableaux of the skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000078The number of alternating sign matrices whose left key is the permutation. St000761The number of ascents in an integer composition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000674The number of hills of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. 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. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition 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. 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. 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. 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000862The number of parts of the shifted shape of a permutation. St000759The smallest missing part in an integer partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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 $eA$. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000091The descent variation of a composition. 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. 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. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks 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. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St000735The last entry on the main diagonal of a standard tableau. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001933The largest multiplicity of a part in an integer partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers.
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