Your data matches 343 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [2] => 2
[[2,2]]
 => [1,2] => [2] => 2
[[1],[2]]
 => [2,1] => [2] => 2
[[1,3]]
 => [1,2] => [2] => 2
[[2,3]]
 => [1,2] => [2] => 2
[[3,3]]
 => [1,2] => [2] => 2
[[1],[3]]
 => [2,1] => [2] => 2
[[2],[3]]
 => [2,1] => [2] => 2
[[1,1,2]]
 => [1,2,3] => [3] => 2
[[1,2,2]]
 => [1,2,3] => [3] => 2
[[2,2,2]]
 => [1,2,3] => [3] => 2
[[1,1],[2]]
 => [3,1,2] => [3] => 2
[[1,2],[2]]
 => [2,1,3] => [3] => 2
[[1,4]]
 => [1,2] => [2] => 2
[[2,4]]
 => [1,2] => [2] => 2
[[3,4]]
 => [1,2] => [2] => 2
[[4,4]]
 => [1,2] => [2] => 2
[[1],[4]]
 => [2,1] => [2] => 2
[[2],[4]]
 => [2,1] => [2] => 2
[[3],[4]]
 => [2,1] => [2] => 2
[[1,1,3]]
 => [1,2,3] => [3] => 2
[[1,2,3]]
 => [1,2,3] => [3] => 2
[[1,3,3]]
 => [1,2,3] => [3] => 2
[[2,2,3]]
 => [1,2,3] => [3] => 2
[[2,3,3]]
 => [1,2,3] => [3] => 2
[[3,3,3]]
 => [1,2,3] => [3] => 2
[[1,1],[3]]
 => [3,1,2] => [3] => 2
[[1,2],[3]]
 => [3,1,2] => [3] => 2
[[1,3],[2]]
 => [2,1,3] => [3] => 2
[[1,3],[3]]
 => [2,1,3] => [3] => 2
[[2,2],[3]]
 => [3,1,2] => [3] => 2
[[2,3],[3]]
 => [2,1,3] => [3] => 2
[[1],[2],[3]]
 => [3,2,1] => [2,1] => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4] => 2
[[1,1,2,2]]
 => [1,2,3,4] => [4] => 2
[[1,2,2,2]]
 => [1,2,3,4] => [4] => 2
[[2,2,2,2]]
 => [1,2,3,4] => [4] => 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => 2
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => 2
[[1,5]]
 => [1,2] => [2] => 2
[[2,5]]
 => [1,2] => [2] => 2
[[3,5]]
 => [1,2] => [2] => 2
[[4,5]]
 => [1,2] => [2] => 2
[[5,5]]
 => [1,2] => [2] => 2
[[1],[5]]
 => [2,1] => [2] => 2
[[2],[5]]
 => [2,1] => [2] => 2
[[3],[5]]
 => [2,1] => [2] => 2
[[4],[5]]
 => [2,1] => [2] => 2
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000388
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => 1 = 2 - 1
[[1,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[5,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St001352
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => 1 = 2 - 1
[[1,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[5,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
Description
The number of internal nodes in the modular decomposition of a graph.
Matching statistic: St001951
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,2]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,3]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,4]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 1 = 2 - 1
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => 1 = 2 - 1
[[1,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[2,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[3,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[4,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[5,5]]
 => [1,2] => [2] => ([],2)
 => 1 = 2 - 1
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => 1 = 2 - 1
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 75% ●values known / values provided: 87%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[2,2]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[1],[2]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[1,3]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[2,3]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[3,3]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[1],[3]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[2],[3]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[1,1,2]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[2,2,2]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,1],[2]]
 => [3,1,2] => [3] => ([],3)
 => 0 = 2 - 2
[[1,2],[2]]
 => [2,1,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,4]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[2,4]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[3,4]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[4,4]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[1],[4]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[2],[4]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[3],[4]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[1,1,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[2,2,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[2,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[3,3,3]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,1],[3]]
 => [3,1,2] => [3] => ([],3)
 => 0 = 2 - 2
[[1,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 0 = 2 - 2
[[1,3],[2]]
 => [2,1,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 0 = 2 - 2
[[2,2],[3]]
 => [3,1,2] => [3] => ([],3)
 => 0 = 2 - 2
[[2,3],[3]]
 => [2,1,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 3 - 2
[[1,1,1,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 0 = 2 - 2
[[1,1,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 0 = 2 - 2
[[1,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 0 = 2 - 2
[[2,2,2,2]]
 => [1,2,3,4] => [4] => ([],4)
 => 0 = 2 - 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => ([],4)
 => 0 = 2 - 2
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => ([],4)
 => 0 = 2 - 2
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => ([],4)
 => 0 = 2 - 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => ([],4)
 => 0 = 2 - 2
[[1,5]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[2,5]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[3,5]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[4,5]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[5,5]]
 => [1,2] => [2] => ([],2)
 => 0 = 2 - 2
[[1],[5]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[2],[5]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[3],[5]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[4],[5]]
 => [2,1] => [2] => ([],2)
 => 0 = 2 - 2
[[1,1,4]]
 => [1,2,3] => [3] => ([],3)
 => 0 = 2 - 2
[[1],[2],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3} - 2
[[1],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3} - 2
[[2],[3],[4]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3} - 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4} - 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4} - 2
[[1],[2],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[1],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[1],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[2],[3],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[2],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[3],[4],[5]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3} - 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4} - 2
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4} - 2
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4} - 2
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4} - 2
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {4,4,4,4} - 2
[[1],[2],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[1],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[1],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[1],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[2],[3],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[2],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[2],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[3],[4],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[3],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[4],[5],[6]]
 => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3} - 2
[[1,2],[2],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,3],[2],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[2],[3]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,4],[2],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[2],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[2],[5]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,3],[3],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,4],[3],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[3],[5]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,4],[4],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[1,5],[4],[5]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[2,3],[3],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[2,4],[3],[5]]
 => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
[[2,5],[3],[4]]
 => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001488
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[2,2]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[1],[2]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[1,3]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[2,3]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[3,3]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[1],[3]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[2],[3]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[1,1,2]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[1,2,2]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[2,2,2]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[1,1],[2]]
 => [3,1,2] => [3] => [[3],[]]
 => 2
[[1,2],[2]]
 => [2,1,3] => [3] => [[3],[]]
 => 2
[[1,4]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[2,4]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[3,4]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[4,4]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[1],[4]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[2],[4]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[3],[4]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[1,1,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[1,2,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[1,3,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[2,2,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[2,3,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[3,3,3]]
 => [1,2,3] => [3] => [[3],[]]
 => 2
[[1,1],[3]]
 => [3,1,2] => [3] => [[3],[]]
 => 2
[[1,2],[3]]
 => [3,1,2] => [3] => [[3],[]]
 => 2
[[1,3],[2]]
 => [2,1,3] => [3] => [[3],[]]
 => 2
[[1,3],[3]]
 => [2,1,3] => [3] => [[3],[]]
 => 2
[[2,2],[3]]
 => [3,1,2] => [3] => [[3],[]]
 => 2
[[2,3],[3]]
 => [2,1,3] => [3] => [[3],[]]
 => 2
[[1],[2],[3]]
 => [3,2,1] => [2,1] => [[2,2],[1]]
 => 3
[[1,1,1,2]]
 => [1,2,3,4] => [4] => [[4],[]]
 => 2
[[1,1,2,2]]
 => [1,2,3,4] => [4] => [[4],[]]
 => 2
[[1,2,2,2]]
 => [1,2,3,4] => [4] => [[4],[]]
 => 2
[[2,2,2,2]]
 => [1,2,3,4] => [4] => [[4],[]]
 => 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4] => [[4],[]]
 => 2
[[1,1,2],[2]]
 => [3,1,2,4] => [4] => [[4],[]]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [4] => [[4],[]]
 => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4] => [[4],[]]
 => 2
[[1,5]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[2,5]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[3,5]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[4,5]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[5,5]]
 => [1,2] => [2] => [[2],[]]
 => 2
[[1],[5]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[2],[5]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[3],[5]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[4],[5]]
 => [2,1] => [2] => [[2],[]]
 => 2
[[1,1,1,1,1,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1,2,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,2,2,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,2,2,2,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,2,2,2,2,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[2,2,2,2,2,2]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1,1],[2]]
 => [6,1,2,3,4,5] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1,2],[2]]
 => [5,1,2,3,4,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,2,2],[2]]
 => [4,1,2,3,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,2,2,2],[2]]
 => [3,1,2,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,2,2,2,2],[2]]
 => [2,1,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[[1,1,1,1,1,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,2,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,2,2,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,2,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,2,2,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,2,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,2,2,2,2,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,2,2,2,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,2,2,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,2,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,3,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[2,2,2,2,2,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[2,2,2,2,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[2,2,2,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[2,2,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[2,3,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[3,3,3,3,3,3]]
 => [1,2,3,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,1],[3]]
 => [6,1,2,3,4,5] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,2],[3]]
 => [6,1,2,3,4,5] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,3],[2]]
 => [5,1,2,3,4,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,1,3],[3]]
 => [5,1,2,3,4,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,2,2],[3]]
 => [6,1,2,3,4,5] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,2,3],[2]]
 => [4,1,2,3,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,2,3],[3]]
 => [5,1,2,3,4,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,3,3],[2]]
 => [4,1,2,3,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,1,3,3],[3]]
 => [4,1,2,3,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,2,2],[3]]
 => [6,1,2,3,4,5] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,2,3],[2]]
 => [3,1,2,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,2,3],[3]]
 => [5,1,2,3,4,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,3,3],[2]]
 => [3,1,2,4,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[1,1,2,3,3],[3]]
 => [4,1,2,3,5,6] => [6] => [[6],[]]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St000771
(load all 7 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2 - 1
[[1,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[4,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[4,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[5,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,4],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,4],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[2,4],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[2,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[3,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,1,3],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,1,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,2,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[2,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[2,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,5],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[3,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[3,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[4,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,1,4],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,1,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,1,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 7 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,2]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[2]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,3]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[3]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,2]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,1],[2]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,2],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2 - 1
[[1,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[4,4]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[4]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[3,3,3]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,1],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[1,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[2,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2,3],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2} - 1
[[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[1,1,1,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,1,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[2,2,2,2]]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 2 - 1
[[1,1,1],[2]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,2],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
[[1,2,2],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2} - 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[2,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[3,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[4,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[5,5]]
 => [1,2] => ([],2)
 => ([],1)
 => 1 = 2 - 1
[[1],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[2],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[3],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[4],[5]]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[[1,1,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,4,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,2,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[2,3,4]]
 => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 2 - 1
[[1,4],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,4],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[2,4],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[2,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[3,4],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2} - 1
[[1,1,3],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,1,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,2,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3,3],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[2,2,3],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[2,3,3],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,2,2,2,2,4,4} - 1
[[1,1,1,2],[2]]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,1,2,2],[2]]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,2,2,2],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2} - 1
[[1,5],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[3]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[2,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[3,5],[4]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[3,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[4,5],[5]]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2} - 1
[[1,1,4],[2]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,1,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,1,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[1,4,4],[4]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,2,4],[3]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,2,4],[4]]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
[[2,3,4],[3]]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4} - 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001198
(load all 5 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,2]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[2]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2
[[1,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[3]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2}
[[2],[3]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2}
[[1,1,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[2]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2
[[1,2],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[4,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[2],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[3],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[1,1,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,2],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,3],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,2],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[2,3],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2}
[[1,1,2],[2]]
 => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2}
[[1,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[4,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[5,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[2],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[3],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[4],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[1,1,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[4,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,2],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,4],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,4],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,2],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2,4],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[3,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[3,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,1,1,3]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,1],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[2,2,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[2,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[2,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[2],[3]]
 => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2],[2],[3]]
 => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {2,2}
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {2,2}
[[1],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[2],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[3],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[4],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[5],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[1,1],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 5 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Values
[[1,2]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,2]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[2]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2
[[1,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,3]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[3]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2}
[[2],[3]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2}
[[1,1,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,2]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[2]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2
[[1,2],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[4,4]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[2],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[3],[4]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2}
[[1,1,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,3,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,2],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,3],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,2],[3]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[2,3],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,3}
[[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,1],[2]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2}
[[1,1,2],[2]]
 => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2}
[[1,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[2,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[3,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[4,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[5,5]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2
[[1],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[2],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[3],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[4],[5]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2}
[[1,1,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,2,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,2,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[2,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,3,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[3,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[4,4,4]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[[1,1],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,2],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,4],[2]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,4],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,2],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2,4],[3]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[2,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[3,3],[4]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[3,4],[4]]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,3,3,3}
[[1,1,1,3]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2
[[1,1,1],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[2,2,2],[3]]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[2,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[2,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1],[2],[3]]
 => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,2],[2],[3]]
 => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,3,4,4}
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {2,2}
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {2,2}
[[1],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[2],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[3],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[4],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[5],[6]]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? ∊ {2,2,2,2,2}
[[1,1],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4],[5]]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
The following 333 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001881The number of factors of a lattice as a Cartesian product of lattices. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000381The largest part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001568The smallest positive integer that does not appear twice in the partition. St001933The largest multiplicity of a part in an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. 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. St001588The number of distinct 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. 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. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000383The last part of an integer composition. St000068The number of minimal elements in a poset. St000527The width of the poset. St000909The number of maximal chains of maximal size in a poset. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000897The number of different multiplicities of parts of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000657The smallest part of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000761The number of ascents in an integer composition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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. St001624The breadth of a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000259The diameter of a connected graph. St000307The number of rowmotion orbits of a poset. St000706The product of the factorials of the multiplicities of an integer partition. St000759The smallest missing part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000993The multiplicity of the largest part of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000143The largest repeated part of a partition. St000475The number of parts equal to 1 in a partition. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000069The number of maximal elements of a poset. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000090The variation of a composition. St000091The descent variation of a composition. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001397Number of pairs of incomparable elements in a finite poset. St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001613The binary logarithm of the size of the center of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000322The skewness of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000096The number of spanning trees of a graph. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000349The number of different adjacency matrices of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000948The chromatic discriminant of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001282The number of graphs with the same chromatic polynomial. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001386The number of prime labellings of a graph. St001463The number of distinct columns in the nullspace of a graph. St001642The Prague dimension of a graph. St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001734The lettericity of a graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001828The Euler characteristic of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000313The number of degree 2 vertices of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001306The number of induced paths on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001692The number of vertices with higher degree than the average degree in a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001621The number of atoms of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000100The number of linear extensions of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001857The number of edges in the reduced word graph of a signed permutation. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St001743The discrepancy of a graph. St000268The number of strongly connected orientations of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000344The number of strongly connected outdegree sequences of a graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001073The number of nowhere zero 3-flows of a graph. St001281The normalized isoperimetric number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St000315The number of isolated vertices of a graph. St001330The hat guessing number of a graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001625The Möbius invariant of a lattice. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000741The Colin de Verdière graph invariant. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000422The energy of a graph, if it is integral. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001409The maximal entry of a semistandard tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000101The cocharge of a semistandard tableau. St001408The number of maximal entries in a semistandard tableau. St001410The minimal entry of a semistandard tableau. St000102The charge of a semistandard tableau. St000327The number of cover relations in a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.