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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St001333
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001333: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
Description
The cardinality of a minimal edge-isolating set of a graph.
Let F be a set of graphs. A set of vertices S is F-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of S does not contain any graph in F.
This statistic returns the cardinality of the smallest isolating set when F contains only the graph with one edge.
Matching statistic: St001393
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001393: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2] => ([],2)
=> 0
[2,1] => [2] => ([],2)
=> 0
[1,2,3] => [3] => ([],3)
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> 1
[2,1,3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => ([],3)
=> 0
[3,1,2] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St001261
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [3] => ([],3)
=> 1 = 0 + 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [4] => ([],4)
=> 1 = 0 + 1
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [4] => ([],4)
=> 1 = 0 + 1
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [4] => ([],4)
=> 1 = 0 + 1
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000480
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ 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)
=> [1,1]
=> 0
[2,1] => [2] => ([],2)
=> [1,1]
=> 0
[1,2,3] => [3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => [3] => ([],3)
=> [1,1,1]
=> 0
[2,3,1] => [3] => ([],3)
=> [1,1,1]
=> 0
[3,1,2] => [3] => ([],3)
=> [1,1,1]
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,3,1,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,3,4,1] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,4,1,3] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,4,1,2] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for n≠2) can cover is
12(√1+8n−3)
and an element which covers this number of elements is given by (c+i,c,c−1,…,3,2,1), where 1≤i≤c+2.
Matching statistic: St001280
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001280: Integer partitions ⟶ 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)
=> [1,1]
=> 0
[2,1] => [2] => ([],2)
=> [1,1]
=> 0
[1,2,3] => [3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
[2,1,3] => [3] => ([],3)
=> [1,1,1]
=> 0
[2,3,1] => [3] => ([],3)
=> [1,1,1]
=> 0
[3,1,2] => [3] => ([],3)
=> [1,1,1]
=> 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,3,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,3,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[2,3,1,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,3,4,1] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,4,1,3] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[3,1,2,4] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,4,1,2] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,1,2,3] => [4] => ([],4)
=> [1,1,1,1]
=> 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3,4,5] => [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000659
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1,0]
=> ? = 0
[1,2] => [2] => [2]
=> [1,0,1,0]
=> 0
[2,1] => [2] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,3,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1,4,2] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1,4] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,4,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,4,1,2] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[4,2,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,1,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St001035
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1,0]
=> ? = 0
[1,2] => [2] => [2]
=> [1,0,1,0]
=> 0
[2,1] => [2] => [2]
=> [1,0,1,0]
=> 0
[1,2,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,3,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,3,1,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1,4,2] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1,4] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,4,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,4,1,2] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[4,2,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,1,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,3,5] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is k-convex if k is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St001418
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [2] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [2] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,3,1] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,2,1] => [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,1,3,4] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,3,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,1,4,2] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,1,3] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,3,4] => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000535
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000535: 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
[1,3,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [3] => [3] => ([],3)
=> 0
[2,3,1] => [3] => [3] => ([],3)
=> 0
[3,1,2] => [3] => [3] => ([],3)
=> 0
[3,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 0
[1,2,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [4] => [4] => ([],4)
=> 0
[2,1,4,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [4] => [4] => ([],4)
=> 0
[2,3,4,1] => [4] => [4] => ([],4)
=> 0
[2,4,1,3] => [4] => [4] => ([],4)
=> 0
[2,4,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[3,1,2,4] => [4] => [4] => ([],4)
=> 0
[3,1,4,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,2,4,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [4] => [4] => ([],4)
=> 0
[3,4,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4] => [4] => ([],4)
=> 0
[4,1,3,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[4,3,1,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,3,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[6,5,7,4,3,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[7,5,6,4,3,2,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The rank-width of a graph.
Matching statistic: St000781
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 98%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 98%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [2] => [2]
=> []
=> ? = 0
[2,1] => [2] => [2]
=> []
=> ? = 0
[1,2,3] => [3] => [3]
=> []
=> ? = 0
[1,3,2] => [1,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [3] => [3]
=> []
=> ? = 0
[2,3,1] => [3] => [3]
=> []
=> ? = 0
[3,1,2] => [3] => [3]
=> []
=> ? = 0
[3,2,1] => [2,1] => [2,1]
=> [1]
=> 1
[1,2,3,4] => [4] => [4]
=> []
=> ? = 0
[1,2,4,3] => [2,2] => [2,2]
=> [2]
=> 1
[1,3,2,4] => [1,3] => [3,1]
=> [1]
=> 1
[1,3,4,2] => [1,3] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [1,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [4] => [4]
=> []
=> ? = 0
[2,1,4,3] => [2,2] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [4] => [4]
=> []
=> ? = 0
[2,3,4,1] => [4] => [4]
=> []
=> ? = 0
[2,4,1,3] => [4] => [4]
=> []
=> ? = 0
[2,4,3,1] => [3,1] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [4] => [4]
=> []
=> ? = 0
[3,1,4,2] => [2,2] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,2] => [2,2]
=> [2]
=> 1
[3,2,4,1] => [2,2] => [2,2]
=> [2]
=> 1
[3,4,1,2] => [4] => [4]
=> []
=> ? = 0
[3,4,2,1] => [3,1] => [3,1]
=> [1]
=> 1
[4,1,2,3] => [4] => [4]
=> []
=> ? = 0
[4,1,3,2] => [3,1] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [2,2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [3,1] => [3,1]
=> [1]
=> 1
[4,3,1,2] => [1,3] => [3,1]
=> [1]
=> 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [5] => [5]
=> []
=> ? = 0
[1,2,3,5,4] => [3,2] => [3,2]
=> [2]
=> 1
[1,2,4,3,5] => [2,3] => [3,2]
=> [2]
=> 1
[1,2,4,5,3] => [2,3] => [3,2]
=> [2]
=> 1
[1,2,5,3,4] => [2,3] => [3,2]
=> [2]
=> 1
[1,2,5,4,3] => [2,2,1] => [2,2,1]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,4] => [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,4] => [4,1]
=> [1]
=> 1
[1,3,4,5,2] => [1,4] => [4,1]
=> [1]
=> 1
[1,3,5,2,4] => [1,4] => [4,1]
=> [1]
=> 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1]
=> 1
[1,4,2,5,3] => [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,2,5] => [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,3,5,2] => [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,2,3] => [1,4] => [4,1]
=> [1]
=> 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,3,4] => [1,4] => [4,1]
=> [1]
=> 1
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,4,5] => [5] => [5]
=> []
=> ? = 0
[2,1,3,5,4] => [3,2] => [3,2]
=> [2]
=> 1
[2,1,4,3,5] => [2,3] => [3,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,3] => [3,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,3] => [3,2]
=> [2]
=> 1
[2,1,5,4,3] => [2,2,1] => [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [5] => [5]
=> []
=> ? = 0
[2,3,1,5,4] => [3,2] => [3,2]
=> [2]
=> 1
[2,3,4,1,5] => [5] => [5]
=> []
=> ? = 0
[2,3,4,5,1] => [5] => [5]
=> []
=> ? = 0
[2,3,5,1,4] => [5] => [5]
=> []
=> ? = 0
[2,3,5,4,1] => [4,1] => [4,1]
=> [1]
=> 1
[2,4,1,3,5] => [5] => [5]
=> []
=> ? = 0
[2,4,1,5,3] => [3,2] => [3,2]
=> [2]
=> 1
[2,4,3,1,5] => [3,2] => [3,2]
=> [2]
=> 1
[2,4,5,1,3] => [5] => [5]
=> []
=> ? = 0
[2,5,1,3,4] => [5] => [5]
=> []
=> ? = 0
[3,1,2,4,5] => [5] => [5]
=> []
=> ? = 0
[3,4,1,2,5] => [5] => [5]
=> []
=> ? = 0
[3,4,5,1,2] => [5] => [5]
=> []
=> ? = 0
[3,5,1,2,4] => [5] => [5]
=> []
=> ? = 0
[4,1,2,3,5] => [5] => [5]
=> []
=> ? = 0
[4,5,1,2,3] => [5] => [5]
=> []
=> ? = 0
[5,1,2,3,4] => [5] => [5]
=> []
=> ? = 0
[1,2,3,4,5,6] => [6] => [6]
=> []
=> ? = 0
[2,1,3,4,5,6] => [6] => [6]
=> []
=> ? = 0
[2,3,1,4,5,6] => [6] => [6]
=> []
=> ? = 0
[2,3,4,1,5,6] => [6] => [6]
=> []
=> ? = 0
[2,3,4,5,1,6] => [6] => [6]
=> []
=> ? = 0
[2,3,4,5,6,1] => [6] => [6]
=> []
=> ? = 0
[2,3,4,6,1,5] => [6] => [6]
=> []
=> ? = 0
[2,3,5,1,4,6] => [6] => [6]
=> []
=> ? = 0
[2,3,5,6,1,4] => [6] => [6]
=> []
=> ? = 0
[2,3,6,1,4,5] => [6] => [6]
=> []
=> ? = 0
[2,4,1,3,5,6] => [6] => [6]
=> []
=> ? = 0
[2,4,5,1,3,6] => [6] => [6]
=> []
=> ? = 0
[2,4,5,6,1,3] => [6] => [6]
=> []
=> ? = 0
[2,4,6,1,3,5] => [6] => [6]
=> []
=> ? = 0
[2,5,1,3,4,6] => [6] => [6]
=> []
=> ? = 0
[2,5,6,1,3,4] => [6] => [6]
=> []
=> ? = 0
[2,6,1,3,4,5] => [6] => [6]
=> []
=> ? = 0
[3,1,2,4,5,6] => [6] => [6]
=> []
=> ? = 0
[3,4,1,2,5,6] => [6] => [6]
=> []
=> ? = 0
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000264The girth of a graph, which is not a tree. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. 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. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000058The order of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St000485The length of the longest cycle of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000862The number of parts of the shifted shape of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001734The lettericity of a graph. St000259The diameter of a connected graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001512The minimum rank of a graph. St001638The book thickness of a graph. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001962The proper pathwidth of a graph. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001859The number of factors of the Stanley symmetric function associated with a permutation. 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. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000454The largest eigenvalue of a graph if it is integral.
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