- FindStat

Identifier

Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤ

Values

[[1]] => [1] => [[1]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1],[1],[1],[1],[1],[1]] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 5

[[2],[1],[1],[1],[1]] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

[[2],[2],[1],[1]] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

[[2],[2],[2]] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2

[[1,1],[1],[1],[1],[1]] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

[[1,1],[1,1],[1],[1]] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

[[1,1],[1,1],[1,1]] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2

[[3],[1],[1],[1]] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

[[3],[2],[1]] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

[[3],[3]] => [3,3] => [[1,2,3],[4,5,6]] => 1

[[2,1],[1],[1],[1]] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

[[2,1],[2],[1]] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

[[2,1],[1,1],[1]] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

[[2,1],[2,1]] => [3,3] => [[1,2,3],[4,5,6]] => 1

[[1,1,1],[1],[1],[1]] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

[[1,1,1],[1,1],[1]] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

[[1,1,1],[1,1,1]] => [3,3] => [[1,2,3],[4,5,6]] => 1

[[4],[1],[1]] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

[[4],[2]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[3,1],[1],[1]] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

[[3,1],[2]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[3,1],[1,1]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[2,2],[1],[1]] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

[[2,2],[2]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[2,2],[1,1]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[2,1,1],[1],[1]] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

[[2,1,1],[2]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[2,1,1],[1,1]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[1,1,1,1],[1],[1]] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

[[1,1,1,1],[1,1]] => [4,2] => [[1,2,5,6],[3,4]] => 1

[[5],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[4,1],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[3,2],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[3,1,1],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[2,2,1],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[2,1,1,1],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

[[1,1,1,1,1],[1]] => [5,1] => [[1,3,4,5,6],[2]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 3 + 2\ q + q^{2}$

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

$F_{5} = 7 + 9\ q + 5\ q^{2} + 2\ q^{3} + q^{4}$

$F_{6} = 11 + 18\ q + 11\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{5}$

$F_{7} = 15 + 30\ q + 22\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$

$F_{8} = 22 + 53\ q + 42\ q^{2} + 24\ q^{3} + 11\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{7}$

$F_{9} = 30 + 85\ q + 78\ q^{2} + 46\ q^{3} + 24\ q^{4} + 11\ q^{5} + 5\ q^{6} + 2\ q^{7} + q^{8}$

$F_{10} = 42 + 139\ q + 138\ q^{2} + 90\ q^{3} + 48\ q^{4} + 24\ q^{5} + 11\ q^{6} + 5\ q^{7} + 2\ q^{8} + q^{9}$

Description

The number of descents of a standard tableau.
Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Map

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau

$T$ labeled down (in English convention) each column to the shape of a partition.

Map

to partition

Description

The underlying integer partition of a plane partition.
This is the partition whose parts are the sums of the individual rows of the plane partition.