- FindStat

Identifier

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001301: Posets ⟶ ℤ

Values

[[1,2]] => ([(0,1)],2) => 0

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

[[1],[2]] => ([],1) => 0

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

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

[[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

[[1,1,2]] => ([(0,1)],2) => 0

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

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

[[1,1],[2]] => ([],1) => 0

[[1,2],[2]] => ([(0,1)],2) => 0

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

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

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

[[3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

[[1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

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

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

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

[[1,1,1,2]] => ([(0,1)],2) => 0

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

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

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

[[1,1,1],[2]] => ([],1) => 0

[[1,1,2],[2]] => ([(0,1)],2) => 0

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

[[1,1],[2,2]] => ([],1) => 0

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

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

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

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

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

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

[[1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

[[1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 0

[[1,4],[4]] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 0

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

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

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

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

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

[[1,1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

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

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

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

[[1,3,3],[2]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

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

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

[[2,2],[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

[[1,1,1,1,2]] => ([(0,1)],2) => 0

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

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

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

[[2,2,2,2,2]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0

[[1,1,1,1],[2]] => ([],1) => 0

[[1,1,1,2],[2]] => ([(0,1)],2) => 0

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

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

[[1,1,1],[2,2]] => ([],1) => 0

[[1,1,2],[2,2]] => ([(0,1)],2) => 0

[[1,6]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0

[[1],[6]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0

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

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

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

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

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

[[1],[4],[5]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

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

[[1,1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

[[1,1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 0

[[1,1,4],[4]] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 0

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

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

[[1,1],[4,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

[[1,3],[2,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 0

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

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

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

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

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

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Description

The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.

Map

subcrystal

Description

The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.