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Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions

Images

[[1]] => [1]

[[2]] => [1]

[[3]] => [1]

[[4]] => [1]

[[5]] => [1]

[[6]] => [1]

[[1,1]] => [2]

[[1,2]] => [1,1]

[[2,2]] => [2]

[[1],[2]] => [1,1]

[[1,3]] => [1,1]

[[2,3]] => [1,1]

[[3,3]] => [2]

[[1],[3]] => [1,1]

[[2],[3]] => [1,1]

[[1,4]] => [1,1]

[[2,4]] => [1,1]

[[3,4]] => [1,1]

[[4,4]] => [2]

[[1],[4]] => [1,1]

[[2],[4]] => [1,1]

[[3],[4]] => [1,1]

[[1,5]] => [1,1]

[[2,5]] => [1,1]

[[3,5]] => [1,1]

[[4,5]] => [1,1]

[[5,5]] => [2]

[[1],[5]] => [1,1]

[[2],[5]] => [1,1]

[[3],[5]] => [1,1]

[[4],[5]] => [1,1]

[[1,6]] => [1,1]

[[2,6]] => [1,1]

[[3,6]] => [1,1]

[[4,6]] => [1,1]

[[5,6]] => [1,1]

[[6,6]] => [2]

[[1],[6]] => [1,1]

[[2],[6]] => [1,1]

[[3],[6]] => [1,1]

[[4],[6]] => [1,1]

[[5],[6]] => [1,1]

[[1,7]] => [1,1]

[[2,7]] => [1,1]

[[3,7]] => [1,1]

[[4,7]] => [1,1]

[[5,7]] => [1,1]

[[6,7]] => [1,1]

[[7,7]] => [2]

[[1],[7]] => [1,1]

[[2],[7]] => [1,1]

[[3],[7]] => [1,1]

[[4],[7]] => [1,1]

[[5],[7]] => [1,1]

[[6],[7]] => [1,1]

[[1,8]] => [1,1]

[[2,8]] => [1,1]

[[3,8]] => [1,1]

[[4,8]] => [1,1]

[[5,8]] => [1,1]

[[6,8]] => [1,1]

[[7,8]] => [1,1]

[[8,8]] => [2]

[[1],[8]] => [1,1]

[[2],[8]] => [1,1]

[[3],[8]] => [1,1]

[[4],[8]] => [1,1]

[[5],[8]] => [1,1]

[[6],[8]] => [1,1]

[[7],[8]] => [1,1]

[[1,1,1]] => [3]

[[1,1,2]] => [2,1]

[[1,2,2]] => [2,1]

[[2,2,2]] => [3]

[[1,1],[2]] => [2,1]

[[1,2],[2]] => [2,1]

[[1,1,3]] => [2,1]

[[1,2,3]] => [1,1,1]

[[1,3,3]] => [2,1]

[[2,2,3]] => [2,1]

[[2,3,3]] => [2,1]

[[3,3,3]] => [3]

[[1,1],[3]] => [2,1]

[[1,2],[3]] => [1,1,1]

[[1,3],[2]] => [1,1,1]

[[1,3],[3]] => [2,1]

[[2,2],[3]] => [2,1]

[[2,3],[3]] => [2,1]

[[1],[2],[3]] => [1,1,1]

[[1,1,4]] => [2,1]

[[1,2,4]] => [1,1,1]

[[1,3,4]] => [1,1,1]

[[1,4,4]] => [2,1]

[[2,2,4]] => [2,1]

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

[[2,4,4]] => [2,1]

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

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

[[4,4,4]] => [3]

[[1,1],[4]] => [2,1]

[[1,2],[4]] => [1,1,1]

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Description

The weight of a semistandard tableau as an integer partition.
The weight (or content) of a semistandard tableaux

$T$ with maximal entry

$m$ is the weak composition

$(\alpha_1, \dots, \alpha_m)$ such that

$\alpha_i$ is the number of letters

$i$ occurring in

$T$ .
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.

Properties

surjective

Sage code

def mapping(T):
    return Partition(sorted(T.weight(), reverse=True))

Weight

Created

Apr 14, 2021 at 10:57 by Martin Rubey

Updated

Apr 14, 2021 at 10:57 by Martin Rubey