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Your data matches 337 different statistics following compositions of up to 3 maps.
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Matching statistic: St000752
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Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 0 = 1 - 1
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St000010
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => [3]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => [3]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [3]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => [4]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => [4]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => [2,2]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => [4]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [4]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => [3,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => [3,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => [3,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => [5]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => [5]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [5]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => [5]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => [5]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => [5]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => [5]
=> 1
Description
The length of the partition.
Matching statistic: St000097
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 1
[1,0,1,0]
=> [2,1] => [2] => 10 => 1
[1,1,0,0]
=> [1,2] => [2] => 10 => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => 101 => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => 100 => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => 100 => 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => 100 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => 1001 => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => 1001 => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => 1010 => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => 1000 => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => 1100 => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => 1000 => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => 1010 => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => 1010 => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => 1000 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => 1000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => 10001 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => 10001 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => 10010 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => 10001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => 10010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => 10100 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => 10000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => 10010 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => 11000 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => 11000 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => 10000 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => 10010 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => 10010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => 10010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => 10010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => 10100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => 10000 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => 11000 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => 10000 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => 11000 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => 11000 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => 10000 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => 10100 => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => 10100 => 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => 10100 => 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => 10000 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => 10000 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => 10000 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => 10000 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000346
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> 1
Description
The number of coarsenings of a partition.
A partition $\mu$ coarsens a partition $\lambda$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2] => [1,1] => 1
[1,1,0,0]
=> [1,2] => [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [1,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => [1,1,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => [1,1,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => [1,1,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => [1,1,2] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => [1,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => [1,1,1,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => [1,1,1,1] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => [1,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => [1,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => [1,1,1,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => [1,1,1,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => [1,1,1,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [1,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => [1,2,1,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => [1,1,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => [1,1,2,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => [2,1,1,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => [2,1,1,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => [1,1,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => [1,2,1,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => [1,1,1,1,1] => 1
Description
The largest part of an integer composition.
Matching statistic: St000533
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [2] => [2]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => [3]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => [3]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [3]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => [4]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => [3,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => [4]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => [2,2]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => [4]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => [4]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => [4]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => [4]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [4]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => [3,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => [3,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => [3,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => [4,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => [5]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => [3,2]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => [3,2]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => [3,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => [5]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => [5]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => [4,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [5]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => [3,2]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => [5]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => [5]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => [5]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => [5]
=> 1
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000808
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2] => [1,1] => 1
[1,1,0,0]
=> [1,2] => [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [1,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => [1,1,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => [1,1,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => [1,1,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => [1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => [1,1,2] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => [1,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => [1,1,1,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => [1,1,1,1] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => [1,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => [1,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => [1,1,1,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => [1,1,1,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => [1,1,1,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => [1,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => [1,1,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => [1,1,2,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => [1,2,1,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => [1,1,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => [1,1,2,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => [2,1,1,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => [2,1,1,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => [1,1,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => [1,1,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => [1,2,1,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => [2,1,1,1] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => [1,2,1,1] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => [1,1,1,1,1] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => [1,1,1,1,1] => 1
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St001330
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [3] => ([],3)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4] => ([],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5] => ([],5)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [5] => ([],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 327 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001486The number of corners of the ribbon associated with an integer composition. St001581The achromatic number of a graph. St001820The size of the image of the pop stack sorting operator. St001623The number of doubly irreducible elements of a lattice. St001846The number of elements which do not have a complement in the lattice. St000982The length of the longest constant subword. St000052The number of valleys of a Dyck path not on the x-axis. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000359The number of occurrences of the pattern 23-1. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St001727The number of invisible inversions of a permutation. St000806The semiperimeter of the associated bargraph. St000053The number of valleys of the Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000292The number of ascents of a binary word. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001651The Frankl number of a lattice. St000442The maximal area to the right of an up step of a Dyck path. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000822The Hadwiger number of the graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000015The number of peaks of a Dyck path. St000039The number of crossings of a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000317The cycle descent number of a permutation. St000356The number of occurrences of the pattern 13-2. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000779The tier of a permutation. St000024The number of double up and double down steps of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000619The number of cyclic descents of a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000105The number of blocks in the set partition. St000167The number of leaves of an ordered tree. St000711The number of big exceedences of a permutation. St000925The number of topologically connected components of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000386The number of factors DDU in a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001726The number of visible inversions of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000522The number of 1-protected nodes of a rooted tree. St000884The number of isolated descents of a permutation. St000647The number of big descents of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000402Half the size of the symmetry class of a permutation. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St000646The number of big ascents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000353The number of inner valleys of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001394The genus of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000035The number of left outer peaks of a permutation. St000846The maximal number of elements covering an element of a poset. St000834The number of right outer peaks of a permutation. St001549The number of restricted non-inversions between exceedances. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000451The length of the longest pattern of the form k 1 2. St001513The number of nested exceedences of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000958The number of Bruhat factorizations of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St000023The number of inner peaks of a permutation. St001470The cyclic holeyness of a permutation. St001731The factorization defect of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St001487The number of inner corners of a skew partition. St000264The girth of a graph, which is not a tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001712The number of natural descents of a standard Young tableau. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000691The number of changes of a binary word. St000847The number of standard Young tableaux whose descent set is the binary word. St000295The length of the border of a binary word. St000260The radius of a connected graph. St000741The Colin de Verdière graph invariant. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001896The number of right descents of a signed permutations. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000253The crossing number of a set partition. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $eA$. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001673The degree of asymmetry of an integer composition. St001884The number of borders of a binary word. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000618The number of self-evacuating tableaux of given shape. St000682The Grundy value of Welter's game on a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St001280The number of parts of an integer partition that are at least two. St001423The number of distinct cubes in a binary word. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001485The modular major index of a binary word. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001730The number of times the path corresponding to a binary word crosses the base line. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. 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. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000237The number of small exceedances. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001115The number of even descents of a permutation. St000456The monochromatic index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000842The breadth of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000153The number of adjacent cycles of a permutation. St001862The number of crossings of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2.
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