- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤ

Values

[2,1] => [[2,2],[1]] => [1] => [1] => 0

[1,2,1] => [[2,2,1],[1]] => [1] => [1] => 0

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

[2,2] => [[3,2],[1]] => [1] => [1] => 0

[3,1] => [[3,3],[2]] => [2] => [2] => 0

[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [1] => 0

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

[1,2,2] => [[3,2,1],[1]] => [1] => [1] => 0

[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] => 1

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

[2,2,1] => [[3,3,2],[2,1]] => [2,1] => [3] => 0

[2,3] => [[4,2],[1]] => [1] => [1] => 0

[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] => 0

[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [1] => 0

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

[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [1] => 0

[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] => 1

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

[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => [3] => 0

[1,2,3] => [[4,2,1],[1]] => [1] => [1] => 0

[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] => 0

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

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

[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [3,1] => 0

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

[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [5] => 0

[2,2,2] => [[4,3,2],[2,1]] => [2,1] => [3] => 0

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

[2,4] => [[5,2],[1]] => [1] => [1] => 0

[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] => 0

[3,3] => [[5,3],[2]] => [2] => [2] => 0

[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [3,2,1] => 0

[4,2] => [[5,4],[3]] => [3] => [2,1] => 0

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

[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [1] => 0

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

[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [1] => 0

[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] => 1

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

[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => [3] => 0

[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [1] => 0

[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] => 0

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

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

[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [3,1] => 0

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

[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [5] => 0

[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => [3] => 0

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

[1,2,4] => [[5,2,1],[1]] => [1] => [1] => 0

[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] => 0

[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] => 0

[1,4,2] => [[5,4,1],[3]] => [3] => [2,1] => 0

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

[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] => 1

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

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

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

[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [5,1] => 0

[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [3,1] => 0

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

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

[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [7] => 0

[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [5] => 0

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

[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [3] => 0

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

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

[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [3,2] => 0

[2,5] => [[6,2],[1]] => [1] => [1] => 0

[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] => 0

[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] => 0

[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [4,1] => 0

[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] => 1

[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3,2,1] => 0

[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [4,3] => 0

[4,3] => [[6,4],[3]] => [3] => [2,1] => 0

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

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

[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] => 0

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].

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.

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.

Map

inner shape

Description

The inner shape of a skew partition.