Processing math: 53%

Your data matches 7 different statistics following compositions of up to 3 maps.
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Mp00248: Permutations DEX compositionInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000454: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,1,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [5] => [5] => ([],5)
=> 0
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,4,1,3,5] => [5] => [5] => ([],5)
=> 0
[2,4,5,1,3] => [5] => [5] => ([],5)
=> 0
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,1,3,4] => [5] => [5] => ([],5)
=> 0
[2,5,1,4,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,3,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[3,1,2,4,5] => [5] => [5] => ([],5)
=> 0
[3,4,1,2,5] => [5] => [5] => ([],5)
=> 0
Description
The largest eigenvalue of a graph if it is integral. If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001270
Mp00248: Permutations DEX compositionInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001270: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,1,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [5] => [5] => ([],5)
=> 0
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,4,1,3,5] => [5] => [5] => ([],5)
=> 0
[2,4,5,1,3] => [5] => [5] => ([],5)
=> 0
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,1,3,4] => [5] => [5] => ([],5)
=> 0
[2,5,1,4,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,3,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[3,1,2,4,5] => [5] => [5] => ([],5)
=> 0
[3,4,1,2,5] => [5] => [5] => ([],5)
=> 0
Description
The bandwidth of a graph. The bandwidth of a graph is the smallest number k such that the vertices of the graph can be ordered as v1,,vn with kd(vi,vj)|ij|. We adopt the convention that the singleton graph has bandwidth 0, consistent with the bandwith of the complete graph on n vertices having bandwidth n1, but in contrast to any path graph on more than one vertex having bandwidth 1. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Mp00248: Permutations DEX compositionInteger compositions
Mp00038: Integer compositions reverseInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001962: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,4,3,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,1,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [5] => [5] => ([],5)
=> 0
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,4,1,3,5] => [5] => [5] => ([],5)
=> 0
[2,4,5,1,3] => [5] => [5] => ([],5)
=> 0
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,1,3,4] => [5] => [5] => ([],5)
=> 0
[2,5,1,4,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,3,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[3,1,2,4,5] => [5] => [5] => ([],5)
=> 0
[3,4,1,2,5] => [5] => [5] => ([],5)
=> 0
Description
The proper pathwidth of a graph. The proper pathwidth ppw(G) was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if G has at least one edge, then ppw(G) is the minimum k for which G is a minor of the Cartesian product KkP of a complete graph on k vertices with a path; and further that ppw(G) is the minor monotone floor Z(G):=min of the [[St000482|zero forcing number]] \operatorname{Z}(G). It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for H in this definition, i.e. \lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}. The minimum degree \delta, treewidth \operatorname{tw}, and pathwidth \operatorname{pw} satisfy \delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1. Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.
Mp00248: Permutations DEX compositionInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001644: Graphs ⟶ ℤResult quality: 74% values known / values provided: 74%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [2] => [2] => ([],2)
=> 0
[2,1] => [2] => [2] => ([],2)
=> 0
[1,2,3] => [3] => [3] => ([],3)
=> 0
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,5,4,3,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,1,4,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,1,5] => [5] => [5] => ([],5)
=> 0
[2,3,4,5,1] => [5] => [5] => ([],5)
=> 0
[2,3,5,1,4] => [5] => [5] => ([],5)
=> 0
[2,3,5,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,4,1,3,5] => [5] => [5] => ([],5)
=> 0
[2,4,5,1,3] => [5] => [5] => ([],5)
=> 0
[2,4,5,3,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,1,3,4] => [5] => [5] => ([],5)
=> 0
[2,5,1,4,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[2,5,3,4,1] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[3,1,2,4,5] => [5] => [5] => ([],5)
=> 0
[3,4,1,2,5] => [5] => [5] => ([],5)
=> 0
[3,4,5,1,2] => [5] => [5] => ([],5)
=> 0
[1,3,2,5,4,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,2,5,6,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,2,6,4,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,6,5,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,2,5,3,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,2,5,6,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,2,6,3,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,3,2,5,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,3,5,2,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,3,5,6,2] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,3,6,2,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,6,5,2,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,2,6,3,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,3,2,4,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,3,6,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,6,2,5,3,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,6,3,2,4,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,6,3,5,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,1,5,2,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,1,5,6,2] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,1,6,2,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,2,1,5,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,2,5,1,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,2,5,6,1] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,2,6,1,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,3,6,5,1,2] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,3,1,6,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,3,2,1,4,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,3,2,6,1,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,1,6,2,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,2,1,3,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,2,6,1,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,3,1,2,6] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,4,3,6,1,2] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,3,1,5,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,3,2,1,4,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,3,2,5,1,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,1,5,2,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,2,1,3,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,2,5,1,3] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,3,1,2,5] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,4,3,5,1,2] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,2,1,3,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[6,5,3,1,2,4] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,2,4,6,5,7] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,3,2,4,6,7,5] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,3,2,4,7,5,6] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,3,4,2,6,5,7] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,3,4,2,6,7,5] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
Description
The dimension of a graph. The dimension of a graph is the least integer n such that there exists a representation of the graph in the Euclidean space of dimension n with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000307
Mp00067: Permutations Foata bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00209: Permutations pattern posetPosets
St000307: Posets ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,3,4,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,5,2,4] => [3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1 + 1
[3,5,2,4,1] => [3,2,5,4,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,1,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[4,3,5,1,2] => [1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[4,5,2,3,1] => [2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[5,1,3,4,2] => [3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[5,2,3,4,1] => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[5,3,1,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[5,3,2,4,1] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[5,3,4,1,2] => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,4,2,1] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,4,1,3,2] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[5,4,2,3,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,4,5,3,6] => [4,1,2,5,3,6] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 2 + 1
[1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,4,6,3,5] => [4,1,2,3,6,5] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,5,6,3,4] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[1,3,2,5,4,6] => [5,3,1,2,4,6] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 3 + 1
[1,3,2,5,6,4] => [5,3,1,2,6,4] => [1,5,6,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(1,9),(2,9),(2,10),(2,12),(3,8),(3,10),(3,11),(4,7),(4,11),(4,12),(5,17),(7,14),(7,15),(8,13),(8,14),(9,13),(9,15),(10,13),(10,16),(11,5),(11,14),(11,16),(12,5),(12,15),(12,16),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 3 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,3,4,6] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,6,1,3,4] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,6,1,3,4,5] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset P. It sends an order ideal I to the order ideal generated by the minimal antichain of P \setminus I.
Matching statistic: St001632
Mp00067: Permutations Foata bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00209: Permutations pattern posetPosets
St001632: Posets ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,3,1] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,4,2,5] => [3,1,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[1,3,4,5,2] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,5,2,4] => [3,1,2,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,4,5,3,2] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[1,5,2,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[1,5,3,4,2] => [5,1,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 3 + 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[2,4,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => [4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,1,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,1,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 1 + 1
[3,5,2,4,1] => [3,2,5,4,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,3,1,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[4,3,5,1,2] => [1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[4,5,1,2,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[4,5,2,3,1] => [2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 + 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 1 + 1
[5,1,3,4,2] => [3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[5,2,3,4,1] => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[5,3,1,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,1,4,2] => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[5,3,2,4,1] => [3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 + 1
[5,3,4,1,2] => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,3,4,2,1] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 + 1
[5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[5,4,1,3,2] => [5,1,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[5,4,2,3,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,4,5,3,6] => [4,1,2,5,3,6] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 2 + 1
[1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,4,6,3,5] => [4,1,2,3,6,5] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,5,6,3,4] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2 + 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[1,3,2,5,4,6] => [5,3,1,2,4,6] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ? = 3 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,4,6,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,5,6,1,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,5,1,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,5,6,1,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,4,6,1,3,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,1,3,4,6] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,5,6,1,3,4] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[2,6,1,3,4,5] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset.
Mp00170: Permutations to signed permutationSigned permutations
Mp00167: Signed permutations inverse Kreweras complementSigned permutations
St001896: Signed permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 60%
Values
[1] => [1] => [-1] => 1 = 0 + 1
[1,2] => [1,2] => [2,-1] => 1 = 0 + 1
[2,1] => [2,1] => [1,-2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,-1] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,3,-2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [1,2,-3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,-2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 1 = 0 + 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 1 = 0 + 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 1 = 0 + 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => ? = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => ? = 2 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => ? = 2 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => ? = 2 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => ? = 2 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => ? = 2 + 1
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => ? = 3 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 1 = 0 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 1 = 0 + 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 1 = 0 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 1 = 0 + 1
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => 1 = 0 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => 2 = 1 + 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => 1 = 0 + 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 1 = 0 + 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => 2 = 1 + 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => 1 = 0 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => 2 = 1 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 2 = 1 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => ? = 0 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 0 + 1
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => ? = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => ? = 1 + 1
[3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => ? = 1 + 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => ? = 1 + 1
[4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => ? = 0 + 1
[4,3,1,2,5] => [4,3,1,2,5] => [4,2,1,5,-3] => ? = 2 + 1
[4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => ? = 2 + 1
[4,3,5,2,1] => [4,3,5,2,1] => [4,2,1,3,-5] => ? = 2 + 1
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [5,4,1,2,-3] => ? = 1 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [3,4,1,2,-5] => ? = 1 + 1
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => ? = 0 + 1
[5,1,2,4,3] => [5,1,2,4,3] => [3,5,4,1,-2] => ? = 1 + 1
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => ? = 1 + 1
[5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,1,-5] => ? = 1 + 1
[5,3,1,2,4] => [5,3,1,2,4] => [4,2,5,1,-3] => ? = 2 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [5,2,4,1,-3] => ? = 2 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,1,-5] => ? = 2 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [5,2,3,1,-4] => ? = 2 + 1
[5,3,4,2,1] => [5,3,4,2,1] => [4,2,3,1,-5] => ? = 2 + 1
[5,4,1,2,3] => [5,4,1,2,3] => [4,5,2,1,-3] => ? = 2 + 1
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => ? = 2 + 1
[5,4,2,3,1] => [5,4,2,3,1] => [3,4,2,1,-5] => ? = 2 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,-1] => ? = 0 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,4,3,5,6,-1] => ? = 2 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [2,5,3,4,6,-1] => ? = 2 + 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [2,6,3,4,5,-1] => ? = 2 + 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [2,5,3,6,4,-1] => ? = 2 + 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [2,4,5,3,6,-1] => ? = 2 + 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [2,5,6,3,4,-1] => ? = 2 + 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [2,4,5,6,3,-1] => ? = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,2,5,4,6,-1] => ? = 3 + 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,2,6,4,5,-1] => ? = 3 + 1
Description
The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.