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Your data matches 334 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
[[2],[]]
=> 0
[[1,1],[]]
=> 0
[[2,1],[1]]
=> 0
[[3],[]]
=> 0
[[2,1],[]]
=> 0
[[3,1],[1]]
=> 0
[[2,2],[1]]
=> 0
[[3,2],[2]]
=> 0
[[1,1,1],[]]
=> 0
[[2,2,1],[1,1]]
=> 0
[[2,1,1],[1]]
=> 0
[[3,2,1],[2,1]]
=> 0
[[4],[]]
=> 0
[[3,1],[]]
=> 0
[[4,1],[1]]
=> 0
[[2,2],[]]
=> 1
[[3,2],[1]]
=> 0
[[4,2],[2]]
=> 0
[[2,1,1],[]]
=> 0
[[3,2,1],[1,1]]
=> 0
[[3,1,1],[1]]
=> 0
[[4,2,1],[2,1]]
=> 0
[[3,3],[2]]
=> 0
[[4,3],[3]]
=> 0
[[2,2,1],[1]]
=> 0
[[3,3,1],[2,1]]
=> 0
[[3,2,1],[2]]
=> 0
[[4,3,1],[3,1]]
=> 0
[[2,2,2],[1,1]]
=> 0
[[3,3,2],[2,2]]
=> 0
[[3,2,2],[2,1]]
=> 0
[[4,3,2],[3,2]]
=> 0
[[1,1,1,1],[]]
=> 0
[[2,2,2,1],[1,1,1]]
=> 0
[[2,2,1,1],[1,1]]
=> 0
[[3,3,2,1],[2,2,1]]
=> 0
[[2,1,1,1],[1]]
=> 0
[[3,2,2,1],[2,1,1]]
=> 0
[[3,2,1,1],[2,1]]
=> 0
[[4,3,2,1],[3,2,1]]
=> 0
[[5],[]]
=> 0
[[4,1],[]]
=> 0
[[5,1],[1]]
=> 0
[[3,2],[]]
=> 1
[[4,2],[1]]
=> 0
[[5,2],[2]]
=> 0
[[3,1,1],[]]
=> 0
[[4,2,1],[1,1]]
=> 0
[[4,1,1],[1]]
=> 0
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
[[2],[]]
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> 0
[[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001305
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(load all 2 compositions to match this statistic)
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[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
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
Description
The number of induced cycles on four vertices in a graph.
Matching statistic: St001311
Values
[[1],[]]
=> ([],1)
=> ([],1)
=> 0
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[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
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
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
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[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
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
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
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[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
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
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
[[2],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[[2,1],[1]]
=> ([],2)
=> ([],2)
=> 0
[[3],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[3,2],[2]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0
[[2,2,1],[1,1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[2,1,1],[1]]
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[[3,2,1],[2,1]]
=> ([],3)
=> ([],3)
=> 0
[[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,2],[2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,2,1],[1,1]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,1,1],[1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[4,2,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[4,3],[3]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,1],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,1],[2]]
=> ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,1],[3,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[3,3,2],[2,2]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,2,2],[2,1]]
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[4,3,2],[3,2]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[[2,2,2,1],[1,1,1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[2,2,1,1],[1,1]]
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[[3,3,2,1],[2,2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[2,1,1,1],[1]]
=> ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[3,2,2,1],[2,1,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[3,2,1,1],[2,1]]
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[[4,3,2,1],[3,2,1]]
=> ([],4)
=> ([],4)
=> 0
[[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,1],[1]]
=> ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[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
[[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[5,2],[2]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[[4,2,1],[1,1]]
=> ([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[[4,1,1],[1]]
=> ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
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: St001064
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[[2],[]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,1],[]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[2,1],[1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3],[]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[2,1],[]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[3,1],[1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[1]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[3,2],[2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[1,1,1],[]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[2,2,1],[1,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[2,1,1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[4],[]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[3,1],[]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[4,1],[1]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2],[]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[3,2],[1]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[4,2],[2]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[2,1,1],[]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[3,1,1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[[4,2,1],[2,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[3,3],[2]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[4,3],[3]]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[2,2,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[3,2,1],[2]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[4,3,1],[3,1]]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[[2,2,2],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[3,3,2],[2,2]]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[5],[]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[4,1],[]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[[5,1],[1]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[[3,2],[]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[4,2],[1]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[5,2],[2]]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[3,1,1],[]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[[4,2,1],[1,1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[[4,1,1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
Matching statistic: St000900
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000900: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[2],[]]
=> []
=> => [1] => 1 = 0 + 1
[[1,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[2,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3],[]]
=> []
=> => [1] => 1 = 0 + 1
[[2,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[3,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[2,2],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3,2],[2]]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[[1,1,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[2,2,1],[1,1]]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[[2,1,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3,2,1],[2,1]]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[[4],[]]
=> []
=> => [1] => 1 = 0 + 1
[[3,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[4,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[2,2],[]]
=> []
=> => [1] => 1 = 0 + 1
[[3,2],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[4,2],[2]]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[[2,1,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[3,2,1],[1,1]]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[[3,1,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[4,2,1],[2,1]]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[[3,3],[2]]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[[4,3],[3]]
=> [3]
=> 1000 => [1,4] => 1 = 0 + 1
[[2,2,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3,3,1],[2,1]]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[[3,2,1],[2]]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[[4,3,1],[3,1]]
=> [3,1]
=> 10010 => [1,3,2] => 1 = 0 + 1
[[2,2,2],[1,1]]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[[3,3,2],[2,2]]
=> [2,2]
=> 1100 => [1,1,3] => 1 = 0 + 1
[[3,2,2],[2,1]]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[[4,3,2],[3,2]]
=> [3,2]
=> 10100 => [1,2,3] => 1 = 0 + 1
[[1,1,1,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1 = 0 + 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 2 = 1 + 1
[[2,1,1,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1 = 0 + 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => [1,2,2] => 1 = 0 + 1
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => [1,2,2,2] => 1 = 0 + 1
[[5],[]]
=> []
=> => [1] => 1 = 0 + 1
[[4,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[5,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[3,2],[]]
=> []
=> => [1] => 1 = 0 + 1
[[4,2],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[5,2],[2]]
=> [2]
=> 100 => [1,3] => 1 = 0 + 1
[[3,1,1],[]]
=> []
=> => [1] => 1 = 0 + 1
[[4,2,1],[1,1]]
=> [1,1]
=> 110 => [1,1,2] => 1 = 0 + 1
[[4,1,1],[1]]
=> [1]
=> 10 => [1,2] => 1 = 0 + 1
[[6,5,3,2,1],[5,3,2,1]]
=> [5,3,2,1]
=> 100101010 => [1,3,2,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[6,5,4,2,1],[5,4,2,1]]
=> [5,4,2,1]
=> 101001010 => [1,2,3,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[6,5,4,3,1],[5,4,3,1]]
=> [5,4,3,1]
=> 101010010 => [1,2,2,3,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[6,5,4,3,2],[5,4,3,2]]
=> [5,4,3,2]
=> 101010100 => [1,2,2,2,3] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[5,5,4,3,2,1],[4,4,3,2,1]]
=> [4,4,3,2,1]
=> 110101010 => [1,1,2,2,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[5,4,4,3,2,1],[4,3,3,2,1]]
=> [4,3,3,2,1]
=> 101101010 => [1,2,1,2,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[5,4,3,3,2,1],[4,3,2,2,1]]
=> [4,3,2,2,1]
=> 101011010 => [1,2,2,1,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[5,4,3,2,2,1],[4,3,2,1,1]]
=> [4,3,2,1,1]
=> 101010110 => [1,2,2,2,1,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
[[6,5,4,3,2,1],[5,4,3,2,1]]
=> [5,4,3,2,1]
=> 1010101010 => [1,2,2,2,2,2] => ? ∊ {0,1,1,1,1,1,1,1,2} + 1
Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000122
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Values
[[1],[]]
=> []
=> []
=> .
=> ? = 0
[[2],[]]
=> []
=> []
=> .
=> ? ∊ {0,0}
[[1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0}
[[2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0}
[[2,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0}
[[3,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0}
[[2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[4],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,0,1}
[[3,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,0,1}
[[4,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[2,2],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,0,1}
[[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[4,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[2,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,0,1}
[[3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[4,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 0
[[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[3,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[4,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0
[[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 0
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[4,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[[1,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,0,1}
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 0
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 0
[[2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 0
[[5],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[4,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[5,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[3,2],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[5,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[3,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[4,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[4,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[5,3],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 0
[[2,2,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[4,2,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[[5,3,1],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0
[[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 0
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[[5,3,2],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[[2,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 0
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 0
[[1,1,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,0,0,1,1,1,1}
[[6],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[5,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[4,2],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[4,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[3,3],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[3,2,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[3,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[2,2,2],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[2,2,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[2,1,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
[[1,1,1,1,1,1],[]]
=> []
=> []
=> .
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2}
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A086581]] counts binary trees avoiding this pattern.
The following 324 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000210Minimum over maximum difference of elements in cycles. St000252The number of nodes of degree 3 of a binary tree. St000317The cycle descent number of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000807The sum of the heights of the valleys of the associated bargraph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001394The genus of a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001271The competition number of a graph. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. 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. St001696The natural major index of a standard Young tableau. 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. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000666The number of right tethers of a permutation. 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. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001651The Frankl number of a lattice. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000768The number of peaks in an integer composition. St001371The length of the longest Yamanouchi prefix of a binary word. St000119The number of occurrences of the pattern 321 in a permutation. St001537The number of cyclic crossings of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000042The number of crossings of a perfect matching. 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. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001047The maximal number of arcs crossing a given arc of a perfect matching. 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. St001175The size of a partition minus the hook length of the base cell. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000547The number of even non-empty partial sums of an integer partition. St000661The number of rises of length 3 of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 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. St001214The aft of an integer partition. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001423The number of distinct cubes in a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. 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. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000761The number of ascents in an integer composition. St000674The number of hills 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. 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. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. 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. St000787The number of flips required to make a perfect matching noncrossing. St000386The number of factors DDU in a Dyck path. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001552The number of inversions between excedances and fixed points of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000622The number of occurrences of the patterns 2143 or 4231 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. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. 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. St000516The number of stretching pairs of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000921The number of internal inversions of a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000655The length of the minimal rise of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. 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. 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. 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. St000478Another weight of a partition according to Alladi. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. 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. 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. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000931The number of occurrences of the pattern UUU in a Dyck path. St000842The breadth of 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. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001964The interval resolution global dimension of a poset. St000764The number of strong records in an integer composition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000805The number of peaks of the associated bargraph. St000902 The minimal number of repetitions of an integer composition. St000078The number of alternating sign matrices whose left key is the permutation. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000862The number of parts of the shifted shape of a permutation. 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001705The number of occurrences of the pattern 2413 in a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. 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. St000570The Edelman-Greene number of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001481The minimal height of a peak of a Dyck path. 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 $eA$. St001871The number of triconnected components of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given 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. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000759The smallest missing part in an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000897The number of different multiplicities of parts of an integer partition. St000475The number of parts equal to 1 in a partition. St000788The number of nesting-similar perfect matchings of a perfect matching. St001490The number of connected components of a skew partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000022The number of fixed points of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000221The number of strong fixed points of a permutation. St001381The fertility of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000056The decomposition (or block) number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000914The sum of the values of the Möbius function of a poset. St001568The smallest positive integer that does not appear twice in the partition. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000284The Plancherel distribution on integer partitions. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000934The 2-degree of an integer 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001487The number of inner corners of a skew partition. St001890The maximum magnitude of the Möbius function of a poset.
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