Identifier
-
Mp00295:
Standard tableaux
—valley composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000145: Integer partitions ⟶ ℤ
Values
[[1]] => [1] => [[1],[]] => [1] => 0
[[1,2]] => [2] => [[2],[]] => [2] => 1
[[1],[2]] => [2] => [[2],[]] => [2] => 1
[[1,2,3]] => [3] => [[3],[]] => [3] => 2
[[1,3],[2]] => [2,1] => [[2,2],[1]] => [2,2] => 0
[[1,2],[3]] => [3] => [[3],[]] => [3] => 2
[[1],[2],[3]] => [3] => [[3],[]] => [3] => 2
[[1,2,3,4]] => [4] => [[4],[]] => [4] => 3
[[1,3,4],[2]] => [2,2] => [[3,2],[1]] => [3,2] => 1
[[1,2,4],[3]] => [3,1] => [[3,3],[2]] => [3,3] => 1
[[1,2,3],[4]] => [4] => [[4],[]] => [4] => 3
[[1,3],[2,4]] => [2,2] => [[3,2],[1]] => [3,2] => 1
[[1,2],[3,4]] => [3,1] => [[3,3],[2]] => [3,3] => 1
[[1,4],[2],[3]] => [3,1] => [[3,3],[2]] => [3,3] => 1
[[1,3],[2],[4]] => [2,2] => [[3,2],[1]] => [3,2] => 1
[[1,2],[3],[4]] => [4] => [[4],[]] => [4] => 3
[[1],[2],[3],[4]] => [4] => [[4],[]] => [4] => 3
[[1,2,3,4,5]] => [5] => [[5],[]] => [5] => 4
[[1,3,4,5],[2]] => [2,3] => [[4,2],[1]] => [4,2] => 2
[[1,2,4,5],[3]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,2,3,5],[4]] => [4,1] => [[4,4],[3]] => [4,4] => 2
[[1,2,3,4],[5]] => [5] => [[5],[]] => [5] => 4
[[1,3,5],[2,4]] => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 0
[[1,2,5],[3,4]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,3,4],[2,5]] => [2,3] => [[4,2],[1]] => [4,2] => 2
[[1,2,4],[3,5]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,2,3],[4,5]] => [4,1] => [[4,4],[3]] => [4,4] => 2
[[1,4,5],[2],[3]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,3,5],[2],[4]] => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 0
[[1,2,5],[3],[4]] => [4,1] => [[4,4],[3]] => [4,4] => 2
[[1,3,4],[2],[5]] => [2,3] => [[4,2],[1]] => [4,2] => 2
[[1,2,4],[3],[5]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,2,3],[4],[5]] => [5] => [[5],[]] => [5] => 4
[[1,4],[2,5],[3]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,3],[2,5],[4]] => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 0
[[1,2],[3,5],[4]] => [4,1] => [[4,4],[3]] => [4,4] => 2
[[1,3],[2,4],[5]] => [2,3] => [[4,2],[1]] => [4,2] => 2
[[1,2],[3,4],[5]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,5],[2],[3],[4]] => [4,1] => [[4,4],[3]] => [4,4] => 2
[[1,4],[2],[3],[5]] => [3,2] => [[4,3],[2]] => [4,3] => 2
[[1,3],[2],[4],[5]] => [2,3] => [[4,2],[1]] => [4,2] => 2
[[1,2],[3],[4],[5]] => [5] => [[5],[]] => [5] => 4
[[1],[2],[3],[4],[5]] => [5] => [[5],[]] => [5] => 4
[[1,2,3,4,5,6]] => [6] => [[6],[]] => [6] => 5
[[1,3,4,5,6],[2]] => [2,4] => [[5,2],[1]] => [5,2] => 3
[[1,2,4,5,6],[3]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,2,3,5,6],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,2,3,4,6],[5]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,2,3,4,5],[6]] => [6] => [[6],[]] => [6] => 5
[[1,3,5,6],[2,4]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5,6],[3,4]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,4,6],[2,5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2,3,6],[4,5]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,4,5],[2,6]] => [2,4] => [[5,2],[1]] => [5,2] => 3
[[1,2,4,5],[3,6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,2,3,5],[4,6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,2,3,4],[5,6]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,4,5,6],[2],[3]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,5,6],[2],[4]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5,6],[3],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,4,6],[2],[5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2,3,6],[4],[5]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,3,4,5],[2],[6]] => [2,4] => [[5,2],[1]] => [5,2] => 3
[[1,2,4,5],[3],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,2,3,5],[4],[6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,2,3,4],[5],[6]] => [6] => [[6],[]] => [6] => 5
[[1,3,5],[2,4,6]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5],[3,4,6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,4],[2,5,6]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2,3],[4,5,6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,6],[2,5],[4]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,6],[3,5],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,6],[2,4],[5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,4,5],[2,6],[3]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,5],[2,6],[4]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5],[3,6],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,4],[2,6],[5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2,3],[4,6],[5]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,3,5],[2,4],[6]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5],[3,4],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,4],[2,5],[6]] => [2,4] => [[5,2],[1]] => [5,2] => 3
[[1,2,4],[3,5],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,2,3],[4,5],[6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,5,6],[2],[3],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,6],[2],[4],[5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2,6],[3],[4],[5]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,4,5],[2],[3],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3,5],[2],[4],[6]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2,5],[3],[4],[6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3,4],[2],[5],[6]] => [2,4] => [[5,2],[1]] => [5,2] => 3
[[1,2,4],[3],[5],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,2,3],[4],[5],[6]] => [6] => [[6],[]] => [6] => 5
[[1,4],[2,5],[3,6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3],[2,5],[4,6]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
[[1,2],[3,5],[4,6]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3],[2,4],[5,6]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,5],[2,6],[3],[4]] => [4,2] => [[5,4],[3]] => [5,4] => 3
[[1,3],[2,6],[4],[5]] => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
[[1,2],[3,6],[4],[5]] => [5,1] => [[5,5],[4]] => [5,5] => 3
[[1,4],[2,5],[3],[6]] => [3,3] => [[5,3],[2]] => [5,3] => 3
[[1,3],[2,5],[4],[6]] => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
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Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
outer shape
Description
The outer shape of the skew partition.
Map
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
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