Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000867: Integer partitions ⟶ ℤ
Values
[1] => [[1],[]] => [1] => [] => 0
[1,1] => [[1,1],[]] => [1,1] => [1] => 1
[2] => [[2],[]] => [2] => [] => 0
[1,1,1] => [[1,1,1],[]] => [1,1,1] => [1,1] => 2
[1,2] => [[2,1],[]] => [2,1] => [1] => 1
[2,1] => [[2,2],[1]] => [2,2] => [2] => 3
[3] => [[3],[]] => [3] => [] => 0
[1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => [1,1,1] => 3
[1,1,2] => [[2,1,1],[]] => [2,1,1] => [1,1] => 2
[1,2,1] => [[2,2,1],[1]] => [2,2,1] => [2,1] => 4
[1,3] => [[3,1],[]] => [3,1] => [1] => 1
[2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => [2,2] => 5
[2,2] => [[3,2],[1]] => [3,2] => [2] => 3
[3,1] => [[3,3],[2]] => [3,3] => [3] => 6
[4] => [[4],[]] => [4] => [] => 0
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => [1,1,1] => 3
[1,1,2,1] => [[2,2,1,1],[1]] => [2,2,1,1] => [2,1,1] => 5
[1,1,3] => [[3,1,1],[]] => [3,1,1] => [1,1] => 2
[1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => [2,2,1] => 6
[1,2,2] => [[3,2,1],[1]] => [3,2,1] => [2,1] => 4
[1,3,1] => [[3,3,1],[2]] => [3,3,1] => [3,1] => 7
[1,4] => [[4,1],[]] => [4,1] => [1] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => [2,2,2] => 7
[2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => [2,2] => 5
[2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => [3,2] => 8
[2,3] => [[4,2],[1]] => [4,2] => [2] => 3
[3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => [3,3] => 9
[3,2] => [[4,3],[2]] => [4,3] => [3] => 6
[4,1] => [[4,4],[3]] => [4,4] => [4] => 10
[5] => [[5],[]] => [5] => [] => 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [1,1,1,1,1] => 5
[1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => [2,1,1,1] => 6
[1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => [1,1,1] => 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => [2,2,1,1] => 7
[1,1,2,2] => [[3,2,1,1],[1]] => [3,2,1,1] => [2,1,1] => 5
[1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => [3,1,1] => 8
[1,1,4] => [[4,1,1],[]] => [4,1,1] => [1,1] => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => [2,2,2,1] => 8
[1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => [2,2,1] => 6
[1,2,2,1] => [[3,3,2,1],[2,1]] => [3,3,2,1] => [3,2,1] => 9
[1,2,3] => [[4,2,1],[1]] => [4,2,1] => [2,1] => 4
[1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => [3,3,1] => 10
[1,3,2] => [[4,3,1],[2]] => [4,3,1] => [3,1] => 7
[1,4,1] => [[4,4,1],[3]] => [4,4,1] => [4,1] => 11
[1,5] => [[5,1],[]] => [5,1] => [1] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => [2,2,2,2] => 9
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => [2,2,2] => 7
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => [3,2,2] => 10
[2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => [2,2] => 5
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => [3,3,2] => 11
[2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => [3,2] => 8
[2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => [4,2] => 12
[2,4] => [[5,2],[1]] => [5,2] => [2] => 3
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => [3,3,3] => 12
[3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => [3,3] => 9
[3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => [4,3] => 13
[3,3] => [[5,3],[2]] => [5,3] => [3] => 6
[4,1,1] => [[4,4,4],[3,3]] => [4,4,4] => [4,4] => 14
[4,2] => [[5,4],[3]] => [5,4] => [4] => 10
[5,1] => [[5,5],[4]] => [5,5] => [5] => 15
[6] => [[6],[]] => [6] => [] => 0
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => [1,1,1,1,1] => 5
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [2,2,1,1,1,1] => [2,1,1,1,1] => 7
[1,1,1,1,3] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [2,2,2,1,1,1] => [2,2,1,1,1] => 8
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [3,2,1,1,1] => [2,1,1,1] => 6
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [3,3,1,1,1] => [3,1,1,1] => 9
[1,1,1,4] => [[4,1,1,1],[]] => [4,1,1,1] => [1,1,1] => 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [2,2,2,2,1,1] => [2,2,2,1,1] => 9
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => [2,2,1,1] => 7
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => [3,2,1,1] => 10
[1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => [2,1,1] => 5
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => [3,3,1,1] => 11
[1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => [3,1,1] => 8
[1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => [4,1,1] => 12
[1,1,5] => [[5,1,1],[]] => [5,1,1] => [1,1] => 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [2,2,2,2,2,1] => [2,2,2,2,1] => 10
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => [2,2,2,1] => 8
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => [3,2,2,1] => 11
[1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => [2,2,1] => 6
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => [3,3,2,1] => 12
[1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => [3,2,1] => 9
[1,2,3,1] => [[4,4,2,1],[3,1]] => [4,4,2,1] => [4,2,1] => 13
[1,2,4] => [[5,2,1],[1]] => [5,2,1] => [2,1] => 4
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [3,3,3,3,1] => [3,3,3,1] => 13
[1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => [3,3,1] => 10
[1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => [4,3,1] => 14
[1,3,3] => [[5,3,1],[2]] => [5,3,1] => [3,1] => 7
[1,4,1,1] => [[4,4,4,1],[3,3]] => [4,4,4,1] => [4,4,1] => 15
[1,4,2] => [[5,4,1],[3]] => [5,4,1] => [4,1] => 11
[1,5,1] => [[5,5,1],[4]] => [5,5,1] => [5,1] => 16
[1,6] => [[6,1],[]] => [6,1] => [1] => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => [2,2,2,2,2] => 11
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [3,2,2,2,2] => [2,2,2,2] => 9
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [3,3,2,2,2] => [3,2,2,2] => 12
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => [2,2,2] => 7
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [3,3,3,2,2] => [3,3,2,2] => 13
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => [3,2,2] => 10
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Description
The sum of the hook lengths in the first row of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
Map
outer shape
Description
The outer shape of the skew 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
first row removal
Description
Removes the first entry of an integer partition
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