Identifier
-
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000764: Integer compositions ⟶ ℤ
Values
[1] => [[1]] => [1] => 1
[2] => [[1,2]] => [2] => 1
[1,1] => [[1],[2]] => [1,1] => 1
[3] => [[1,2,3]] => [3] => 1
[2,1] => [[1,3],[2]] => [1,2] => 2
[1,1,1] => [[1],[2],[3]] => [1,1,1] => 1
[4] => [[1,2,3,4]] => [4] => 1
[3,1] => [[1,3,4],[2]] => [1,3] => 2
[2,2] => [[1,2],[3,4]] => [2,2] => 1
[2,1,1] => [[1,4],[2],[3]] => [1,1,2] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [1,1,1,1] => 1
[5] => [[1,2,3,4,5]] => [5] => 1
[4,1] => [[1,3,4,5],[2]] => [1,4] => 2
[3,2] => [[1,2,5],[3,4]] => [2,3] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [1,1,3] => 2
[2,2,1] => [[1,3],[2,5],[4]] => [1,2,2] => 2
[2,1,1,1] => [[1,5],[2],[3],[4]] => [1,1,1,2] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [1,1,1,1,1] => 1
[6] => [[1,2,3,4,5,6]] => [6] => 1
[5,1] => [[1,3,4,5,6],[2]] => [1,5] => 2
[4,2] => [[1,2,5,6],[3,4]] => [2,4] => 2
[4,1,1] => [[1,4,5,6],[2],[3]] => [1,1,4] => 2
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => 1
[3,2,1] => [[1,3,6],[2,5],[4]] => [1,2,3] => 3
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [1,1,1,3] => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => 1
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [1,1,2,2] => 2
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [1,1,1,1,2] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [1,1,1,1,1,1] => 1
[7] => [[1,2,3,4,5,6,7]] => [7] => 1
[6,1] => [[1,3,4,5,6,7],[2]] => [1,6] => 2
[5,2] => [[1,2,5,6,7],[3,4]] => [2,5] => 2
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [1,1,5] => 2
[4,3] => [[1,2,3,7],[4,5,6]] => [3,4] => 2
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [1,2,4] => 3
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [1,1,1,4] => 2
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [1,3,3] => 2
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [2,2,3] => 2
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [1,1,2,3] => 3
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [1,1,1,1,3] => 2
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [1,2,2,2] => 2
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [1,1,1,2,2] => 2
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [1,1,1,1,1,2] => 2
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,1] => 1
[8] => [[1,2,3,4,5,6,7,8]] => [8] => 1
[7,1] => [[1,3,4,5,6,7,8],[2]] => [1,7] => 2
[6,2] => [[1,2,5,6,7,8],[3,4]] => [2,6] => 2
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [1,1,6] => 2
[5,3] => [[1,2,3,7,8],[4,5,6]] => [3,5] => 2
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [1,2,5] => 3
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [1,1,1,5] => 2
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => 1
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [1,3,4] => 3
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [2,2,4] => 2
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [1,1,2,4] => 3
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [1,1,1,1,4] => 2
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [2,3,3] => 2
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [1,1,3,3] => 2
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [1,2,2,3] => 3
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [1,1,1,2,3] => 3
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [1,1,1,1,1,3] => 2
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => 1
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [1,1,2,2,2] => 2
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,2] => 2
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,2] => 2
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,1] => 1
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => 1
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [1,8] => 2
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [2,7] => 2
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [1,1,7] => 2
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [3,6] => 2
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [1,2,6] => 3
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [1,1,1,6] => 2
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [4,5] => 2
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [1,3,5] => 3
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [2,2,5] => 2
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [1,1,2,5] => 3
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [1,1,1,1,5] => 2
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [1,4,4] => 2
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [2,3,4] => 3
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [1,1,3,4] => 3
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [1,2,2,4] => 3
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [1,1,1,2,4] => 3
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [1,1,1,1,1,4] => 2
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => 1
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [1,2,3,3] => 3
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [1,1,1,3,3] => 2
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [2,2,2,3] => 2
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [1,1,2,2,3] => 3
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [1,1,1,1,2,3] => 3
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [1,1,1,1,1,1,3] => 2
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [1,2,2,2,2] => 2
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [1,1,1,2,2,2] => 2
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [1,1,1,1,1,2,2] => 2
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [1,1,1,1,1,1,1,2] => 2
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [1,1,1,1,1,1,1,1,1] => 1
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [1,9] => 2
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5] => 1
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [1,1,4,4] => 2
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [2,2,3,3] => 2
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [1,1,1,1,3,3] => 2
>>> Load all 110 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
Description
The number of strong records in an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
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 $i$.
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 $i$.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!