- FindStat

Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000277: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,4,5],[3]] => [2,3] => 9

[[1,2,3,5],[4]] => [3,2] => 9

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

[[1,3,5],[2,4]] => [1,2,2] => 16

[[1,2,5],[3,4]] => [2,3] => 9

[[1,3,4],[2,5]] => [1,3,1] => 11

[[1,2,4],[3,5]] => [2,2,1] => 16

[[1,2,3],[4,5]] => [3,2] => 9

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

[[1,3,5],[2],[4]] => [1,2,2] => 16

[[1,2,5],[3],[4]] => [2,1,2] => 11

[[1,3,4],[2],[5]] => [1,3,1] => 11

[[1,2,4],[3],[5]] => [2,2,1] => 16

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

[[1,4],[2,5],[3]] => [1,1,2,1] => 9

[[1,3],[2,5],[4]] => [1,2,2] => 16

[[1,2],[3,5],[4]] => [2,1,2] => 11

[[1,3],[2,4],[5]] => [1,2,1,1] => 9

[[1,2],[3,4],[5]] => [2,2,1] => 16

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

[[1,4],[2],[3],[5]] => [1,1,2,1] => 9

[[1,3],[2],[4],[5]] => [1,2,1,1] => 9

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

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

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

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

[[1,2,4,5,6],[3]] => [2,4] => 14

[[1,2,3,5,6],[4]] => [3,3] => 19

[[1,2,3,4,6],[5]] => [4,2] => 14

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

[[1,3,5,6],[2,4]] => [1,2,3] => 35

[[1,2,5,6],[3,4]] => [2,4] => 14

[[1,3,4,6],[2,5]] => [1,3,2] => 40

[[1,2,4,6],[3,5]] => [2,2,2] => 61

[[1,2,3,6],[4,5]] => [3,3] => 19

[[1,3,4,5],[2,6]] => [1,4,1] => 19

[[1,2,4,5],[3,6]] => [2,3,1] => 40

[[1,2,3,5],[4,6]] => [3,2,1] => 35

[[1,2,3,4],[5,6]] => [4,2] => 14

[[1,4,5,6],[2],[3]] => [1,1,4] => 10

[[1,3,5,6],[2],[4]] => [1,2,3] => 35

[[1,2,5,6],[3],[4]] => [2,1,3] => 26

[[1,3,4,6],[2],[5]] => [1,3,2] => 40

[[1,2,4,6],[3],[5]] => [2,2,2] => 61

[[1,2,3,6],[4],[5]] => [3,1,2] => 26

[[1,3,4,5],[2],[6]] => [1,4,1] => 19

[[1,2,4,5],[3],[6]] => [2,3,1] => 40

[[1,2,3,5],[4],[6]] => [3,2,1] => 35

[[1,2,3,4],[5],[6]] => [4,1,1] => 10

[[1,3,5],[2,4,6]] => [1,2,2,1] => 61

[[1,2,5],[3,4,6]] => [2,3,1] => 40

[[1,3,4],[2,5,6]] => [1,3,2] => 40

[[1,2,4],[3,5,6]] => [2,2,2] => 61

[[1,2,3],[4,5,6]] => [3,3] => 19

[[1,4,6],[2,5],[3]] => [1,1,2,2] => 35

[[1,3,6],[2,5],[4]] => [1,2,3] => 35

[[1,2,6],[3,5],[4]] => [2,1,3] => 26

[[1,3,6],[2,4],[5]] => [1,2,1,2] => 40

[[1,2,6],[3,4],[5]] => [2,2,2] => 61

[[1,4,5],[2,6],[3]] => [1,1,3,1] => 26

[[1,3,5],[2,6],[4]] => [1,2,2,1] => 61

[[1,2,5],[3,6],[4]] => [2,1,2,1] => 40

[[1,3,4],[2,6],[5]] => [1,3,2] => 40

[[1,2,4],[3,6],[5]] => [2,2,2] => 61

[[1,2,3],[4,6],[5]] => [3,1,2] => 26

[[1,3,5],[2,4],[6]] => [1,2,2,1] => 61

[[1,2,5],[3,4],[6]] => [2,3,1] => 40

[[1,3,4],[2,5],[6]] => [1,3,1,1] => 26

[[1,2,4],[3,5],[6]] => [2,2,1,1] => 35

[[1,2,3],[4,5],[6]] => [3,2,1] => 35

[[1,5,6],[2],[3],[4]] => [1,1,1,3] => 10

[[1,4,6],[2],[3],[5]] => [1,1,2,2] => 35

[[1,3,6],[2],[4],[5]] => [1,2,1,2] => 40

[[1,2,6],[3],[4],[5]] => [2,1,1,2] => 19

[[1,4,5],[2],[3],[6]] => [1,1,3,1] => 26

[[1,3,5],[2],[4],[6]] => [1,2,2,1] => 61

[[1,2,5],[3],[4],[6]] => [2,1,2,1] => 40

[[1,3,4],[2],[5],[6]] => [1,3,1,1] => 26

[[1,2,4],[3],[5],[6]] => [2,2,1,1] => 35

[[1,2,3],[4],[5],[6]] => [3,1,1,1] => 10

[[1,4],[2,5],[3,6]] => [1,1,2,2] => 35

[[1,3],[2,5],[4,6]] => [1,2,2,1] => 61

>>> Load all 1724 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.