- FindStat

Your data matches 385 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000157
(load all 110 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [[1]]
 => 0
[1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1
[1,1,0,0]
 => [1,2] => [[1,2]]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [[1],[2],[3]]
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [[1,3],[2]]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[1,3],[2]]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[1,2],[3],[4]]
 => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[1,3],[2],[4]]
 => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[1,3],[2],[4]]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[1,4],[2],[3]]
 => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[1,4],[2],[3]]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[1,2,3],[4],[5]]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[1,2],[3,4],[5]]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[1,2,4],[3],[5]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[1,3,4],[2],[5]]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[1,3,4],[2],[5]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[1,3,4],[2],[5]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[1,2],[3,5],[4]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[1,3],[2,5],[4]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[1,3],[2,5],[4]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[1,2],[3,5],[4]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[1,2,5],[3],[4]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[1,3,5],[2],[4]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[1,4,5],[2],[3]]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[1,4,5],[2],[3]]
 => 2

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St001644
(load all 3 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,2] => ([],2)
 => ([],1)
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[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)
 => 0
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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)
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[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)
 => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1
[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)
 => 2
[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)
 => 2
[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)
 => 1
[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)
 => 0
[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)
 => 4
[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)
 => 3
[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)
 => 3
[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)
 => 2
[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)
 => 3
[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)
 => 2
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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)
 => 2
[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)
 => 2
[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)
 => 1
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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)
 => 2
[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)
 => 1
[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)
 => 2
[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)
 => 3
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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)
 => 2
[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)
 => 2
[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)
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2

Description

The dimension of a graph. The dimension of a graph is the least integer

$n$ such that there exists a representation of the graph in the Euclidean space of dimension

$n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

Matching statistic: St000010
(load all 26 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,0]
 => [2] => [2]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => [3]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[]
 => [] => ?
 => ? = 0 + 1

Description

The length of the partition.

Matching statistic: St000097
(load all 16 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [2] => ([],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[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)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[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)
 => 5 = 4 + 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)
 => 4 = 3 + 1
[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)
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[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)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[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)
 => 4 = 3 + 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)
 => 3 = 2 + 1
[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)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[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)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[]
 => [] => ?
 => ? = 0 + 1

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000288
(load all 120 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => 1 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => 11 => 2 = 1 + 1
[1,1,0,0]
 => [2] => 10 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => 110 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => 101 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => 101 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => 100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => 1101 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => 1011 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => 1011 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => 1010 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => 1001 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => 1001 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 11011 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 11010 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 11001 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 11001 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 10110 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 10111 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 10111 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 10110 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 10101 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 10100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 10010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => 10011 => 3 = 2 + 1
[]
 => [] => ? => ? = 0 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000377

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1]
 => [1]
 => 0
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => [1,1,1]
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [3]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [3]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [3]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => [1,1,1,1]
 => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [2,1,1]
 => 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [2,2]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [2,2]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [2,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [2,2]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [2,2]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [2,2]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [2,2]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [3,1]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => [1,1,1,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [2,1,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [2,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [4,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [4,1]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [4,1]
 => 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,4,2,3,1] => ?
 => ?
 => ? = 5

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St000394
(load all 9 compositions to match this statistic)

Mapping

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1,0]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[]
 => []
 => [] => ?
 => ? = 0

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St000098
(load all 15 compositions to match this statistic)

Mapping

Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [2] => ([],2)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [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)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [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)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [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)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [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)
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [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)
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [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)
 => 3 = 2 + 1
[]
 => []
 => [] => ?
 => ? = 0 + 1

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000147
(load all 9 compositions to match this statistic)

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1]
 => [1]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,1,0,0]
 => [2] => [2]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [3]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [2,1,1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => [3,1,1]
 => 3 = 2 + 1
[]
 => [] => ?
 => ?
 => ? = 0 + 1

Description

The largest part of an integer partition.

Matching statistic: St000228

Mapping

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [2] => [2]
 => 2 = 1 + 1
[1,1,0,0]
 => [2] => [1] => [1]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [3]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => [1] => [1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [4]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2] => [2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2] => [2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1] => [1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [5]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => [3,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => [2,1,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => [2,1,1]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,1,2] => [2,1,1]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => [3,1]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,3] => [3,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,3] => [3,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,1] => [2,1]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => [2,1]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,2] => [2,1]
 => 3 = 2 + 1
[]
 => [] => ? => ?
 => ? = 0 + 1

Description

The size of a partition. This statistic is the constant statistic of the level sets.

The following 375 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001389The number of partitions of the same length below the given integer partition. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001176The size of a partition minus its first part. St000172The Grundy number of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000011The number of touch points (or returns) of a Dyck path. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000822The Hadwiger number of the graph. St000012The area of a Dyck path. St000984The number of boxes below precisely one peak. St000013The height of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000676The number of odd rises of a Dyck path. St001058The breadth of the ordered tree. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000521The number of distinct subtrees of an ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St001651The Frankl number of a lattice. St001812The biclique partition number of a graph. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000439The position of the first down step of a Dyck path. St000741The Colin de Verdière graph invariant. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000925The number of topologically connected components of a set partition. St000167The number of leaves of an ordered tree. St000024The number of double up and double down steps of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001777The number of weak descents in an integer composition. St000025The number of initial rises of a Dyck path. St000273The domination number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000544The cop number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000808The number of up steps of the associated bargraph. St000916The packing number of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001809The index of the step at the first peak of maximal height in a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000527The width of the poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001829The common independence number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000308The height of the tree associated to a permutation. St000071The number of maximal chains in a poset. St001875The number of simple modules with projective dimension at most 1. St000068The number of minimal elements in a poset. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000662The staircase size of the code of a permutation. St000632The jump number of the poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000306The bounce count of a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000225Difference between largest and smallest parts in a partition. St001128The exponens consonantiae of a partition. St000245The number of ascents of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000868The aid statistic in the sense of Shareshian-Wachs. St000159The number of distinct parts of the integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000052The number of valleys of a Dyck path not on the x-axis. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000522The number of 1-protected nodes of a rooted tree. St000164The number of short pairs. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) 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. St000619The number of cyclic descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St000161The sum of the sizes of the right subtrees of a binary tree. St000069The number of maximal elements of a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000141The maximum drop size of a permutation. St000246The number of non-inversions of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000451The length of the longest pattern of the form k 1 2. St000764The number of strong records in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000646The number of big ascents of a permutation. St000214The number of adjacencies of a permutation. St001427The number of descents of a signed permutation. St000153The number of adjacent cycles of a permutation. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000332The positive inversions of an alternating sign matrix. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000843The decomposition number of a perfect matching. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001645The pebbling number of a connected graph. St000356The number of occurrences of the pattern 13-2. St000553The number of blocks of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000065The number of entries equal to -1 in an alternating sign matrix. St000989The number of final rises of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000204The number of internal nodes of a binary tree. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000742The number of big ascents of a permutation after prepending zero. St000991The number of right-to-left minima of a permutation. 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. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000331The number of upper interactions of a Dyck path. St000386The number of factors DDU in a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001153The number of blocks with even minimum in a set partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000084The number of subtrees. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000201The number of leaf nodes in a binary tree. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000485The length of the longest cycle of a permutation. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000740The last entry of a permutation. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001280The number of parts of an integer partition that are at least two. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001828The Euler characteristic of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000094The depth of an ordered tree. St000301The number of facets of the stable set polytope 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. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000216The absolute length of a permutation. St000061The number of nodes on the left branch of a binary tree. St001480The number of simple summands of the module J^2/J^3. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001323The independence gap of a graph. St000142The number of even parts of a partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000312The number of leaves in a graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000710The number of big deficiencies of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St000146The Andrews-Garvan crank of a partition. St000143The largest repeated part of a partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000711The number of big exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001487The number of inner corners of a skew partition. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001430The number of positive entries in a signed permutation. St001152The number of pairs with even minimum in a perfect matching. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000317The cycle descent number of a permutation. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St001668The number of points of the poset minus the width of the poset. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001820The size of the image of the pop stack sorting operator. St001864The number of excedances of a signed permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001769The reflection length of a signed permutation. St001896The number of right descents of a signed permutations. St001866The nesting alignments of a signed permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St001874Lusztig's a-function for the symmetric group. St001964The interval resolution global dimension of a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001343The dimension of the reduced incidence algebra of a poset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .