Your data matches 136 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001271
(load all 11 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ 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]
 => [1,2] => ([],2)
 => 0
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 0
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
Description
The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St001276
(load all 2 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 0
[1,0,1,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St001316
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001316: Graphs ⟶ 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]
 => [2,1] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
Description
The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001733
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001733: Dyck paths ⟶ 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
[1,0,1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [2,1] => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
Matching statistic: St000273
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,2] => ([],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [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)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,5,6,7,2,3,4] => [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,2,3,4,5] => [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => [4,3,1,7,6,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,6,7,1,3,4,5] => [5,4,3,1,7,6,2] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,6,1,2,7] => [7,2,1,6,5,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,7,1,2,6] => [6,2,1,7,5,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => [5,2,1,7,6,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => [4,2,1,7,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => [5,4,2,1,7,6,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,1,5,6,7,2,3] => [3,2,7,6,5,1,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => [3,7,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => [3,2,7,1,6,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => [6,3,2,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => [5,3,2,1,7,6,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => [4,3,2,7,6,1,5] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,1,2,3,7,4] => [4,7,3,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => [4,3,2,7,1,6,5] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
Description
The domination number of a graph. The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St001829
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,2] => ([],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [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)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,5,6,7,2,3,4] => [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,2,3,4,5] => [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => [4,3,1,7,6,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,6,7,1,3,4,5] => [5,4,3,1,7,6,2] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,6,1,2,7] => [7,2,1,6,5,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,4,5,7,1,2,6] => [6,2,1,7,5,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => [5,2,1,7,6,4,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => [4,2,1,7,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => [5,4,2,1,7,6,3] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,1,5,6,7,2,3] => [3,2,7,6,5,1,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => [3,7,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => [3,2,7,1,6,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => [6,3,2,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => [5,3,2,1,7,6,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => [4,3,2,7,6,1,5] => ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,1,2,3,7,4] => [4,7,3,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => [4,3,2,7,1,6,5] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
Description
The common independence number of a graph. The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St000658
(load all 5 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1}
[1,0,1,1,0,0]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1}
[1,1,0,0,1,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[1,1,0,1,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1}
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1}
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,0,0,0,1,0]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,0,0,1,0,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,0,1,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1}
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1}
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [5,4,2,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [4,4,2,2,1]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => [5,2,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of rises of length 2 of a Dyck path. This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St001122
(load all 15 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 1
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => []
 => ? ∊ {0,0}
[1,1,1,0,0,0]
 => [3,1,2] => [3]
 => []
 => ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,1,1}
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2]
 => [2]
 => 0
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,1]
 => [1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [4,1]
 => [1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,1]
 => [1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,1]
 => [1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [4,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [3,2]
 => [2]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [4,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [3,2]
 => [2]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,1,2,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,5,6,2,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,5,1,2,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,1,6] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,1,7,5] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,1,6,7,4] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,5,1,7,4,6] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,5,6,7,1,4] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,6,1,4,7,5] => [7]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001440
(load all 5 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => 0
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? = 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => 0
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => []
 => ? ∊ {0,1}
[1,1,1,0,0,0]
 => [3,1,2] => [3]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,1,1,1}
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4]
 => []
 => ? ∊ {0,1,1,1}
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,2]
 => [2]
 => 0
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [4]
 => []
 => ? ∊ {0,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [4,1]
 => [1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,1]
 => [1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,1]
 => [1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,2]
 => [2]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [4,1]
 => [1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,2]
 => [2]
 => 0
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [3,2]
 => [2]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [4,1]
 => [1]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [3,2]
 => [2]
 => 0
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,1,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,4,6,2,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,1,5,2,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,1,2,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,5,6,2,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,5,1,2,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,1,2,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,1,6,2,3,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,1,6] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,1,7,5] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,1,6,7,4] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,5,1,7,4,6] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,5,6,7,1,4] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,6,1,4,7,5] => [7]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St001124
(load all 6 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,1}
[1,1,0,0]
 => [2,1] => [2]
 => []
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,1,1}
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,1,1}
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,1,1}
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => [2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1}
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1}
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => [2]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => [2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => [2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => [2]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [3,1,1]
 => [1,1]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [3,2]
 => [2]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => 0
[1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [4,1,1]
 => [1,1]
 => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,4,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,3,6,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,1,6] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,4,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,5,3,1,6] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,3,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,5,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,4,6,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,4,3,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,5,6,4,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [3,4,2,5,1,6] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,2,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,2,6,5,1] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,2,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => [5,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,4,2,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 126 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000906The length of the shortest maximal chain in a poset. St000674The number of hills of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001587Half of the largest even part of an integer partition. St000137The Grundy value of an integer partition. St001657The number of twos in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St000260The radius of a connected graph. St001339The irredundance number of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001172The number of 1-rises at odd height of a Dyck path. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St000908The length of the shortest maximal antichain in a poset. St000929The constant term of the character polynomial of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000990The first ascent of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000661The number of rises of length 3 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001568The smallest positive integer that does not appear twice in the partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001128The exponens consonantiae of a partition. St001175The size of a partition minus the hook length of the base cell. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000310The minimal degree of a vertex of a graph. St000456The monochromatic index of a connected graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001481The minimal height of a peak of a Dyck path. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St001691The number of kings in a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000648The number of 2-excedences of a permutation. St000546The number of global descents of a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000733The row containing the largest entry of a standard tableau. St001061The number of indices that are both descents and recoils of a permutation. St000007The number of saliances of the permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000873The aix statistic of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001728The number of invisible descents of a permutation. St000352The Elizalde-Pak rank of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000054The first entry of the permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001820The size of the image of the pop stack sorting operator. St000850The number of 1/2-balanced pairs in a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000764The number of strong records in an integer composition. St001964The interval resolution global dimension of a poset. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St001895The oddness of a signed permutation. St001399The distinguishing number of a poset. St001330The hat guessing number of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001557The number of inversions of the second entry of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001621The number of atoms of a lattice.