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Your data matches 228 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => 01 => 0
[1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => 01 => 0
[1,1,1,0,0,0]
=> [3,1,2] => 10 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 010 => 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 010 => 0
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 100 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0100 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0001 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0010 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 0101 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 0010 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 0100 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000390
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => 0 => 0
[1,1,0,0]
=> [1,2] => 1 => 1
[1,0,1,0,1,0]
=> [3,2,1] => 00 => 0
[1,0,1,1,0,0]
=> [2,3,1] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => 00 => 0
[1,1,0,1,0,0]
=> [2,1,3] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 001 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 011 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0000 => 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 0001 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 0001 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0000 => 0
Description
The number of runs of ones in a binary word.
Matching statistic: St001271
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000745
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => [[1,2]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [[1,2],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000481
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> [2]
=> 0
[1,1,0,0]
=> [1,2] => ([],2)
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 1
[1,1,0,0]
=> [1,2] => [2,1] => 1 => 0
[1,0,1,0,1,0]
=> [2,3,1] => [2,1,3] => 10 => 0
[1,0,1,1,0,0]
=> [2,1,3] => [2,3,1] => 01 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => 10 => 0
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,2,1,4] => 110 => 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => 101 => 0
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => 010 => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 0
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,4,2,1] => 011 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,1,3] => 110 => 0
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 101 => 0
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 101 => 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,4,3,1] => 011 => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,3,1,2] => 110 => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [4,1,3,2] => 101 => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => 0110 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => 0101 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => 1010 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 1101 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => 1011 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => 0110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => 0101 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1011 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 0
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000929
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [1,1]
=> [2]
=> 0
[1,1,0,0]
=> [2] => [2]
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [2,1]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [3]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [3]
=> 0
[1,1,0,1,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [3,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [2,2]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [2,2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [4]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [2,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,2,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [4,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [3,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,2,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [5]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [5]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [4,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [5]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [2,1,1,1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [4,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000932
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000948
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000948: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000948: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([],2)
=> ([],2)
=> 0
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> ([],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
Description
The chromatic discriminant of a graph.
The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$.
According to [1], it equals the cardinality of any of the following sets:
(1) Acyclic orientations of G with unique sink at $q$,
(2) Maximum $G$-parking functions relative to $q$,
(3) Minimal $q$-critical states,
(4) Spanning trees of G without broken circuits,
(5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$,
(6) Multilinear Lyndon heaps on $G$.
In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.
Matching statistic: St000986
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
The following 218 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000733The row containing the largest entry of a standard tableau. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000504The cardinality of the first block of a set partition. St000971The smallest closer of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000989The number of final rises of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000734The last entry in the first row of a standard tableau. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000475The number of parts equal to 1 in a partition. St000352The Elizalde-Pak rank of a permutation. St000315The number of isolated vertices of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000234The number of global ascents of a permutation. St000069The number of maximal elements of a poset. St000884The number of isolated descents of a permutation. St000007The number of saliances of the permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St000068The number of minimal elements in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. 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. St000237The number of small exceedances. St000661The number of rises of length 3 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. 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. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000546The number of global descents of a permutation. St000883The number of longest increasing subsequences of a permutation. St000260The radius of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000617The number of global maxima of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000153The number of adjacent cycles of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000480The number of lower covers of a partition in dominance order. St001022Number of simple modules with projective dimension 3 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. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000740The last entry of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St000658The number of rises of length 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001593This is the number of standard Young tableaux of the given shifted shape. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000054The first entry of the permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001095The number of non-isomorphic posets with precisely one further covering relation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000944The 3-degree of an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000456The monochromatic index of a connected graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001172The number of 1-rises at odd height of a Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001691The number of kings in a graph. St001826The maximal number of leaves on a vertex of a graph. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000917The open packing number of a graph. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001672The restrained domination number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000552The number of cut vertices of a graph. St001948The number of augmented double ascents of a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000215The number of adjacencies of a permutation, zero appended. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000534The number of 2-rises of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000461The rix statistic of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000754The Grundy value for the game of removing nestings in a perfect matching. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000663The number of right floats of a permutation. St001403The number of vertical separators in a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St000366The number of double descents of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000221The number of strong fixed points of a permutation. St000338The number of pixed points of a permutation. St000873The aix statistic of a permutation. St000648The number of 2-excedences of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000296The length of the symmetric border of a binary word. St000756The sum of the positions of the left to right maxima of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001820The size of the image of the pop stack sorting operator. St000850The number of 1/2-balanced pairs in a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001964The interval resolution global dimension of a poset. St000894The trace of an alternating sign matrix. St001846The number of elements which do not have a complement in the lattice. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001330The hat guessing number of a graph. St000214The number of adjacencies of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000649The number of 3-excedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000741The Colin de Verdière graph invariant. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001050The number of terminal closers of a set partition.
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