- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St001611: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,1,3]] => [2,1] => [1,1,1] => 5

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

[[1,3,3]] => [2,1] => [1,1,1] => 5

[[2,2,3]] => [2,1] => [1,1,1] => 5

[[2,3,3]] => [2,1] => [1,1,1] => 5

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

[[1,1],[3]] => [2,1] => [1,1,1] => 5

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

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

[[1,3],[3]] => [2,1] => [1,1,1] => 5

[[2,2],[3]] => [2,1] => [1,1,1] => 5

[[2,3],[3]] => [2,1] => [1,1,1] => 5

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

[[1,1,1,2]] => [3,1] => [1,1,1,1] => 15

[[1,1,2,2]] => [2,2] => [2,1,1] => 11

[[1,2,2,2]] => [3,1] => [1,1,1,1] => 15

[[2,2,2,2]] => [4] => [2,2] => 9

[[1,1,1],[2]] => [3,1] => [1,1,1,1] => 15

[[1,1,2],[2]] => [2,2] => [2,1,1] => 11

[[1,2,2],[2]] => [3,1] => [1,1,1,1] => 15

[[1,1],[2,2]] => [2,2] => [2,1,1] => 11

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

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

[[3,5]] => [1,1] => [2] => 2

[[4,5]] => [1,1] => [2] => 2

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

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

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

[[3],[5]] => [1,1] => [2] => 2

[[4],[5]] => [1,1] => [2] => 2

[[1,1,4]] => [2,1] => [1,1,1] => 5

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

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

[[1,4,4]] => [2,1] => [1,1,1] => 5

[[2,2,4]] => [2,1] => [1,1,1] => 5

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

[[2,4,4]] => [2,1] => [1,1,1] => 5

[[3,3,4]] => [2,1] => [1,1,1] => 5

[[3,4,4]] => [2,1] => [1,1,1] => 5

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

[[1,1],[4]] => [2,1] => [1,1,1] => 5

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

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

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

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

[[1,4],[4]] => [2,1] => [1,1,1] => 5

[[2,2],[4]] => [2,1] => [1,1,1] => 5

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

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

[[2,4],[4]] => [2,1] => [1,1,1] => 5

[[3,3],[4]] => [2,1] => [1,1,1] => 5

[[3,4],[4]] => [2,1] => [1,1,1] => 5

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

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

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

[[1,1,1,3]] => [3,1] => [1,1,1,1] => 15

[[1,1,2,3]] => [2,1,1] => [3,1] => 7

[[1,1,3,3]] => [2,2] => [2,1,1] => 11

[[1,2,2,3]] => [2,1,1] => [3,1] => 7

[[1,2,3,3]] => [2,1,1] => [3,1] => 7

[[1,3,3,3]] => [3,1] => [1,1,1,1] => 15

[[2,2,2,3]] => [3,1] => [1,1,1,1] => 15

[[2,2,3,3]] => [2,2] => [2,1,1] => 11

[[2,3,3,3]] => [3,1] => [1,1,1,1] => 15

[[3,3,3,3]] => [4] => [2,2] => 9

[[1,1,1],[3]] => [3,1] => [1,1,1,1] => 15

[[1,1,2],[3]] => [2,1,1] => [3,1] => 7

[[1,1,3],[2]] => [2,1,1] => [3,1] => 7

[[1,1,3],[3]] => [2,2] => [2,1,1] => 11

[[1,2,2],[3]] => [2,1,1] => [3,1] => 7

[[1,2,3],[2]] => [2,1,1] => [3,1] => 7

[[1,2,3],[3]] => [2,1,1] => [3,1] => 7

[[1,3,3],[2]] => [2,1,1] => [3,1] => 7

[[1,3,3],[3]] => [3,1] => [1,1,1,1] => 15

[[2,2,2],[3]] => [3,1] => [1,1,1,1] => 15

[[2,2,3],[3]] => [2,2] => [2,1,1] => 11

[[2,3,3],[3]] => [3,1] => [1,1,1,1] => 15

[[1,1],[2,3]] => [2,1,1] => [3,1] => 7

[[1,1],[3,3]] => [2,2] => [2,1,1] => 11

[[1,2],[2,3]] => [2,1,1] => [3,1] => 7

[[1,2],[3,3]] => [2,1,1] => [3,1] => 7

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$F_{(2, 2)} = 3\ q^{2}$

$F_{(2, 3)} = 5\ q^{2}$

$F_{(3, 2)} = q^{4} + 4\ q^{5}$

$F_{(2, 4)} = 7\ q^{2}$

$F_{(3, 3)} = 4\ q^{3} + q^{4} + 8\ q^{5}$

$F_{(4, 2)} = q^{9} + 3\ q^{11} + 4\ q^{15}$

$F_{(2, 5)} = 9\ q^{2}$

$F_{(3, 4)} = 12\ q^{3} + q^{4} + 12\ q^{5}$

$F_{(4, 3)} = 15\ q^{7} + q^{9} + 6\ q^{11} + 8\ q^{15}$

$F_{(5, 2)} = q^{16} + 4\ q^{21} + 6\ q^{52}$

$F_{(2, 6)} = 11\ q^{2}$

$F_{(3, 5)} = 24\ q^{3} + q^{4} + 16\ q^{5}$

$F_{(4, 4)} = 10\ q^{5} + 45\ q^{7} + q^{9} + 9\ q^{11} + 12\ q^{15}$

$F_{(5, 3)} = q^{16} + 8\ q^{21} + 21\ q^{26} + 15\ q^{36} + 12\ q^{52}$

$F_{(6, 2)} = q^{31} + 4\ q^{52} + 4\ q^{74} + 6\ q^{135}$

$F_{(2, 7)} = 13\ q^{2}$

$F_{(3, 6)} = 40\ q^{3} + q^{4} + 20\ q^{5}$

$F_{(4, 5)} = 40\ q^{5} + 90\ q^{7} + q^{9} + 12\ q^{11} + 16\ q^{15}$

$F_{(5, 4)} = 52\ q^{12} + q^{16} + 12\ q^{21} + 63\ q^{26} + 45\ q^{36} + 18\ q^{52}$

$F_{(6, 3)} = q^{31} + 8\ q^{52} + 11\ q^{66} + 8\ q^{74} + 15\ q^{92} + 12\ q^{135} + 48\ q^{203}$

$F_{(7, 2)} = q^{57} + 4\ q^{109} + 6\ q^{198} + 8\ q^{371}$

$F_{(2, 8)} = 15\ q^{2}$

$F_{(3, 7)} = 60\ q^{3} + q^{4} + 24\ q^{5}$

$F_{(4, 6)} = 100\ q^{5} + 150\ q^{7} + q^{9} + 15\ q^{11} + 20\ q^{15}$

$F_{(5, 5)} = 26\ q^{7} + 208\ q^{12} + q^{16} + 16\ q^{21} + 126\ q^{26} + 90\ q^{36} + 24\ q^{52}$

$F_{(6, 4)} = 120\ q^{29} + q^{31} + 56\ q^{38} + 12\ q^{52} + 33\ q^{66} + 12\ q^{74} + 45\ q^{92} + 18\ q^{135} + 144\ q^{203}$

$F_{(7, 3)} = q^{57} + 15\ q^{98} + 8\ q^{109} + 12\ q^{198} + 39\ q^{296} + 16\ q^{371} + 30\ q^{566} + 48\ q^{877}$

$F_{(8, 2)} = q^{109} + 4\ q^{212} + 5\ q^{392} + 6\ q^{444} + 8\ q^{1075}$

Description

The number of multiset partitions such that the multiplicities of elements are given by a partition.
In particular, the value on the partition

$(n)$ is the number of integer partitions of

$n$ , oeis:A000041, whereas the value on the partition

$(1^n)$ is the number of set partitions oeis:A006110.

Map

Loehr-Warrington inverse

Description

Return a partition whose length is the diagonal inversion number of the preimage.

Map

weight

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.