Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => [1] => 1
[1,2,1] => [[2,2,1],[1]] => [1] => [1] => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [1,1] => 2
[2,2] => [[3,2],[1]] => [1] => [1] => 1
[3,1] => [[3,3],[2]] => [2] => [2] => 0
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [1] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [1,1] => 2
[1,2,2] => [[3,2,1],[1]] => [1] => [1] => 1
[1,3,1] => [[3,3,1],[2]] => [2] => [2] => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => [1,1] => 2
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => [3] => 1
[2,3] => [[4,2],[1]] => [1] => [1] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [4] => 0
[3,2] => [[4,3],[2]] => [2] => [2] => 0
[4,1] => [[4,4],[3]] => [3] => [2,1] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [1] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [1,1] => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [1] => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => [2] => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => [1,1] => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => [3] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => [1] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [4] => 0
[1,3,2] => [[4,3,1],[2]] => [2] => [2] => 0
[1,4,1] => [[4,4,1],[3]] => [3] => [2,1] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1,1] => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [3,1] => 2
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => [1,1] => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [5] => 1
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => [3] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => [2,1,1] => 2
[2,4] => [[5,2],[1]] => [1] => [1] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [6] => 0
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [4] => 0
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [4,1] => 1
[3,3] => [[5,3],[2]] => [2] => [2] => 0
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [3,2,1] => 2
[4,2] => [[5,4],[3]] => [3] => [2,1] => 1
[5,1] => [[5,5],[4]] => [4] => [2,2] => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [1] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [1,1] => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [1] => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => [2] => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => [1,1] => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => [3] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [1] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [4] => 0
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => [2] => 0
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => [2,1] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1,1] => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [3,1] => 2
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => [1,1] => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [5] => 1
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => [3] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => [2,1,1] => 2
[1,2,4] => [[5,2,1],[1]] => [1] => [1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [6] => 0
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [4] => 0
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [4,1] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => [2] => 0
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [3,2,1] => 2
[1,4,2] => [[5,4,1],[3]] => [3] => [2,1] => 1
[1,5,1] => [[5,5,1],[4]] => [4] => [2,2] => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1,1] => 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1,1] => 4
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [3,1,1] => 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1,1] => 3
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [5,1] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [3,1] => 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [2,1,1,1] => 3
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => [1,1] => 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [7] => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [5] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [3,3] => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [3] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [3,2,1,1] => 3
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => [2,1,1] => 2
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [3,2] => 1
[2,5] => [[6,2],[1]] => [1] => [1] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [8] => 0
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [6] => 0
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [6,1] => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [4] => 0
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [5,2,1] => 2
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [4,1] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [4,2] => 0
[3,4] => [[6,3],[2]] => [2] => [2] => 0
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [3,3,2,1] => 3
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3,2,1] => 2
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [4,3] => 1
[4,3] => [[6,4],[3]] => [3] => [2,1] => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [4,4] => 0
[5,2] => [[6,5],[4]] => [4] => [2,2] => 0
[6,1] => [[6,6],[5]] => [5] => [2,2,1] => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => [1] => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [1,1] => 2
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Description
The number of odd parts of a partition.
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
2-conjugate
Description
Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.
Map
inner shape
Description
The inner shape of a skew partition.
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