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Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
Description
The height of a poset. This equals the rank of the poset [[St000080]] plus one.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of maximal antichains in a poset.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
Description
The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
Description
The rank of the poset.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 3 - 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 3 - 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 3 - 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 3 - 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 2 = 3 - 1
Description
The number of elements in the poset.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[[3,2,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,1],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,3,1],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[6,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,2,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,2,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[6,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,2,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,2,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,5,1],[4,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,4,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,1,1],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,3,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,3,1],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[3,2,1,1,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[5,4,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[4,4,4,1],[3,3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
The following 535 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001616The number of neutral elements in a lattice. St001617The dimension of the space of valuations of a lattice. St001626The number of maximal proper sublattices of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001668The number of points of the poset minus the width of the poset. St001720The minimal length of a chain of small intervals in a lattice. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001820The size of the image of the pop stack sorting operator. St001651The Frankl number of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000299The number of nonisomorphic vertex-induced subtrees. St000384The maximal part of the shifted composition of an integer partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000553The number of blocks of a graph. St000668The least common multiple of the parts of the partition. St000680The Grundy value for Hackendot on posets. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000906The length of the shortest maximal chain in a poset. St001093The detour number of a graph. St001279The sum of the parts of an integer partition that are at least two. St001286The annihilation number of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000258The burning number of a graph. St000271The chromatic index of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000327The number of cover relations in a poset. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000636The hull number of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000907The number of maximal antichains of minimal length in a poset. St000918The 2-limited packing number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001494The Alon-Tarsi number of a graph. St001512The minimum rank of a graph. St001581The achromatic number of a graph. St001613The binary logarithm of the size of the center of a lattice. St001619The number of non-isomorphic sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001655The general position number of a graph. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001717The largest size of an interval in a poset. St001734The lettericity of a graph. 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. St001881The number of factors of a lattice as a Cartesian product of lattices. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000535The rank-width of a graph. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000913The number of ways to refine the partition into singletons. St001057The Grundy value of the game of creating an independent set in a graph. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001623The number of doubly irreducible elements of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001716The 1-improper chromatic number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. 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. 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. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001846The number of elements which do not have a complement in the lattice. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000142The number of even parts of a partition. St000151The number of facets in the chain polytope of the poset. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000313The number of degree 2 vertices of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001092The number of distinct even parts of a partition. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-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. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001572The minimal number of edges to remove to make a graph bipartite. 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. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001689The number of celebrities in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the TurĆ”n graph. St001964The interval resolution global dimension of a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001754The number of tolerances of a finite lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Mƶbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000741The Colin de VerdiĆØre graph invariant. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000480The number of lower covers of a partition in dominance order. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St000897The number of different multiplicities of parts of an integer partition. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nāˆ’1}]$ by adding $c_0$ to $c_{nāˆ’1}$. St000287The number of connected components of a graph. St001271The competition number of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000296The length of the symmetric border of a binary word. St000914The sum of the values of the Mƶbius function of a poset. St001272The number of graphs with the same degree sequence. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001324The minimal number of occurrences of the chordal-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. St001367The smallest number which does not occur as degree of a vertex in a graph. St001696The natural major index of a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001890The maximum magnitude of the Mƶbius function of a poset. St000402Half the size of the symmetry class of a permutation. St000842The breadth of a permutation. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000651The maximal size of a rise in a permutation. St000742The number of big ascents of a permutation after prepending zero. St000057The Shynar inversion number of a standard tableau. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. 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. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001335The cardinality of a minimal cycle-isolating set of a graph. St000312The number of leaves in a graph. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000286The number of connected components of the complement of a graph. St001162The minimum jump of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001344The neighbouring number of a permutation. St001665The number of pure excedances of a permutation. St000095The number of triangles of a graph. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000989The number of final rises of a permutation. St001381The fertility of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. 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. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001871The number of triconnected components of a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000768The number of peaks in an integer composition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St000255The number of reduced Kogan faces with the permutation as type. St000733The row containing the largest entry of a standard tableau. St000758The length of the longest staircase fitting into an integer composition. St000640The rank of the largest boolean interval in a poset. St000652The maximal difference between successive positions of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St000042The number of crossings of a perfect matching. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000761The number of ascents in an integer composition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001198The 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 $eA$. St001206The 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 $eA$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001577The minimal number of edges to add or remove to make a graph a cograph. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000542The number of left-to-right-minima of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000260The radius of a connected graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000273The domination number of a graph. St000349The number of different adjacency matrices of a graph. St000387The matching number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St000948The chromatic discriminant of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001342The number of vertices in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001368The number of vertices of maximal degree in a graph. 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. 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). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001546The number of monomials in the Tutte polynomial of a graph. St001672The restrained domination number of a graph. St001694The number of maximal dissociation sets in a graph. St001737The number of descents of type 2 in a permutation. St001765The number of connected components of the friends and strangers 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. St001829The common independence number of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001917The order of toric promotion on the set of labellings of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000317The cycle descent number of a permutation. St000368The Altshuler-Steinberg determinant 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. St000674The number of hills of a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001306The number of induced paths on four vertices 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. St001319The minimal number of occurrences of the star-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. 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. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001351The Albertson index of a graph. St001356The number of vertices in prime modules of a graph. St001374The Padmakar-Ivan index of a graph. St001479The number of bridges of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001638The book thickness of a graph. 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. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001703The villainy of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001708The number of pairs of vertices of different degree in a graph. St001736The total number of cycles in 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. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001799The number of proper separations of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001826The maximal number of leaves on a vertex of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000694The number of affine bounded permutations that project to a given permutation. St001118The acyclic chromatic index of a graph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001281The normalized isoperimetric number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001592The maximal number of simple paths between any two different vertices of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000379The number of Hamiltonian cycles in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001481The minimal height of a peak of a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001060The distinguishing index of a graph. St001108The 2-dynamic chromatic number of a graph. 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. St001260The permanent of an alternating sign matrix. St001570The minimal number of edges to add to make a graph Hamiltonian. St000929The constant term of the character polynomial of an integer partition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000508Eigenvalues of the random-to-random operator acting on a simple module. 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. St000862The number of parts of the shifted shape of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000007The number of saliances of the permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001484The number of singletons of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001730The number of times the path corresponding to a binary word crosses the base line. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000422The energy of a graph, if it is integral. St000822The Hadwiger number of the graph. St000069The number of maximal elements of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001081The number of minimal length factorizations of a permutation into star transpositions. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St001061The number of indices that are both descents and recoils of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001520The number of strict 3-descents. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.