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Your data matches 55 different statistics following compositions of up to 3 maps.
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Matching statistic: St000745
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
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%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 1
[1,2] => [[1,2]]
=> 1
[2,1] => [[1],[2]]
=> 2
[1,2,3] => [[1,2,3]]
=> 1
[1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [[1,3],[2]]
=> 2
[2,3,1] => [[1,2],[3]]
=> 1
[3,1,2] => [[1,3],[2]]
=> 2
[3,2,1] => [[1],[2],[3]]
=> 3
[1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,3,2] => [[1,2],[3],[4]]
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> 1
[2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [[1,2],[3,4]]
=> 1
[2,4,3,1] => [[1,2],[3],[4]]
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,2,4,1] => [[1,3],[2],[4]]
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,4,2,1] => [[1,2],[3],[4]]
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> 3
[4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> 3
[4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1
[1,2] => [2] => 10 => 1
[2,1] => [1,1] => 11 => 2
[1,2,3] => [3] => 100 => 1
[1,3,2] => [2,1] => 101 => 1
[2,1,3] => [1,2] => 110 => 2
[2,3,1] => [2,1] => 101 => 1
[3,1,2] => [1,2] => 110 => 2
[3,2,1] => [1,1,1] => 111 => 3
[1,2,3,4] => [4] => 1000 => 1
[1,2,4,3] => [3,1] => 1001 => 1
[1,3,2,4] => [2,2] => 1010 => 1
[1,3,4,2] => [3,1] => 1001 => 1
[1,4,2,3] => [2,2] => 1010 => 1
[1,4,3,2] => [2,1,1] => 1011 => 1
[2,1,3,4] => [1,3] => 1100 => 2
[2,1,4,3] => [1,2,1] => 1101 => 2
[2,3,1,4] => [2,2] => 1010 => 1
[2,3,4,1] => [3,1] => 1001 => 1
[2,4,1,3] => [2,2] => 1010 => 1
[2,4,3,1] => [2,1,1] => 1011 => 1
[3,1,2,4] => [1,3] => 1100 => 2
[3,1,4,2] => [1,2,1] => 1101 => 2
[3,2,1,4] => [1,1,2] => 1110 => 3
[3,2,4,1] => [1,2,1] => 1101 => 2
[3,4,1,2] => [2,2] => 1010 => 1
[3,4,2,1] => [2,1,1] => 1011 => 1
[4,1,2,3] => [1,3] => 1100 => 2
[4,1,3,2] => [1,2,1] => 1101 => 2
[4,2,1,3] => [1,1,2] => 1110 => 3
[4,2,3,1] => [1,2,1] => 1101 => 2
[4,3,1,2] => [1,1,2] => 1110 => 3
[4,3,2,1] => [1,1,1,1] => 1111 => 4
[1,2,3,4,5] => [5] => 10000 => 1
[1,2,3,5,4] => [4,1] => 10001 => 1
[1,2,4,3,5] => [3,2] => 10010 => 1
[1,2,4,5,3] => [4,1] => 10001 => 1
[1,2,5,3,4] => [3,2] => 10010 => 1
[1,2,5,4,3] => [3,1,1] => 10011 => 1
[1,3,2,4,5] => [2,3] => 10100 => 1
[1,3,2,5,4] => [2,2,1] => 10101 => 1
[1,3,4,2,5] => [3,2] => 10010 => 1
[1,3,4,5,2] => [4,1] => 10001 => 1
[1,3,5,2,4] => [3,2] => 10010 => 1
[1,3,5,4,2] => [3,1,1] => 10011 => 1
[1,4,2,3,5] => [2,3] => 10100 => 1
[1,4,2,5,3] => [2,2,1] => 10101 => 1
[1,4,3,2,5] => [2,1,2] => 10110 => 1
[1,4,3,5,2] => [2,2,1] => 10101 => 1
[1,4,5,2,3] => [3,2] => 10010 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000363
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
Description
The number of minimal vertex covers of a graph.
A '''vertex cover''' of a graph G is a subset S of the vertices of G such that each edge of G contains at least one vertex of S. A vertex cover is minimal if it contains the least possible number of vertices.
This is also the leading coefficient of the clique polynomial of the complement of G.
This is also the number of independent sets of maximal cardinality of G.
Matching statistic: St000382
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [2] => 2
[1,2,3] => [3,2,1] => [1,1,1] => 1
[1,3,2] => [3,1,2] => [1,2] => 1
[2,1,3] => [2,3,1] => [2,1] => 2
[2,3,1] => [2,1,3] => [1,2] => 1
[3,1,2] => [1,3,2] => [2,1] => 2
[3,2,1] => [1,2,3] => [3] => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,2,4,3] => [4,3,1,2] => [1,1,2] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1
[1,3,4,2] => [4,2,1,3] => [1,1,2] => 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => 1
[1,4,3,2] => [4,1,2,3] => [1,3] => 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => 2
[2,1,4,3] => [3,4,1,2] => [2,2] => 2
[2,3,1,4] => [3,2,4,1] => [1,2,1] => 1
[2,3,4,1] => [3,2,1,4] => [1,1,2] => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1
[2,4,3,1] => [3,1,2,4] => [1,3] => 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => 2
[3,1,4,2] => [2,4,1,3] => [2,2] => 2
[3,2,1,4] => [2,3,4,1] => [3,1] => 3
[3,2,4,1] => [2,3,1,4] => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1
[3,4,2,1] => [2,1,3,4] => [1,3] => 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 2
[4,1,3,2] => [1,4,2,3] => [2,2] => 2
[4,2,1,3] => [1,3,4,2] => [3,1] => 3
[4,2,3,1] => [1,3,2,4] => [2,2] => 2
[4,3,1,2] => [1,2,4,3] => [3,1] => 3
[4,3,2,1] => [1,2,3,4] => [4] => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [2] => 2
[1,2,3] => [3,2,1] => [1,1,1] => 1
[1,3,2] => [2,3,1] => [2,1] => 1
[2,1,3] => [3,1,2] => [1,2] => 2
[2,3,1] => [1,3,2] => [2,1] => 1
[3,1,2] => [2,1,3] => [1,2] => 2
[3,2,1] => [1,2,3] => [3] => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => 1
[1,4,3,2] => [2,3,4,1] => [3,1] => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => 2
[2,1,4,3] => [3,4,1,2] => [2,2] => 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1
[2,4,3,1] => [1,3,4,2] => [3,1] => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => 2
[3,1,4,2] => [2,4,1,3] => [2,2] => 2
[3,2,1,4] => [4,1,2,3] => [1,3] => 3
[3,2,4,1] => [1,4,2,3] => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1
[3,4,2,1] => [1,2,4,3] => [3,1] => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => 2
[4,1,3,2] => [2,3,1,4] => [2,2] => 2
[4,2,1,3] => [3,1,2,4] => [1,3] => 3
[4,2,3,1] => [1,3,2,4] => [2,2] => 2
[4,3,1,2] => [2,1,3,4] => [1,3] => 3
[4,3,2,1] => [1,2,3,4] => [4] => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000025
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of D.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000759
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> []
=> 1
[1,2] => [2] => [1,1,0,0]
=> []
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [1]
=> 2
[1,2,3] => [3] => [1,1,1,0,0,0]
=> []
=> 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St001050
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> {{1}}
=> {{1}}
=> 1
[1,2] => [[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 1
[2,1] => [[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 2
[1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
[2,4,3,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[3,2,4,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
[3,4,2,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[4,2,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St001135
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,2] => [2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,1,4,3] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[3,2,4,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[4,2,3,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001733The number of weak left to right maxima of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000234The number of global ascents of a permutation. St000667The greatest common divisor of the parts of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001571The Cartan determinant of the integer partition. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000989The number of final rises of a permutation. St000617The number of global maxima of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima 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. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000546The number of global descents of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000454The largest eigenvalue of a graph if it is integral.
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