- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001604: Integer partitions ⟶ ℤ

Values

000 => [3] => [3] => [1,1,1] => 0

001 => [2,1] => [2,1] => [3] => 1

010 => [1,1,1] => [1,1,1] => [2,1] => 0

011 => [1,2] => [2,1] => [3] => 1

100 => [1,2] => [2,1] => [3] => 1

101 => [1,1,1] => [1,1,1] => [2,1] => 0

110 => [2,1] => [2,1] => [3] => 1

111 => [3] => [3] => [1,1,1] => 0

0000 => [4] => [4] => [1,1,1,1] => 0

0001 => [3,1] => [3,1] => [2,1,1] => 0

0010 => [2,1,1] => [2,1,1] => [2,2] => 1

0011 => [2,2] => [2,2] => [4] => 1

0100 => [1,1,2] => [2,1,1] => [2,2] => 1

0101 => [1,1,1,1] => [1,1,1,1] => [3,1] => 0

0110 => [1,2,1] => [2,1,1] => [2,2] => 1

0111 => [1,3] => [3,1] => [2,1,1] => 0

1000 => [1,3] => [3,1] => [2,1,1] => 0

1001 => [1,2,1] => [2,1,1] => [2,2] => 1

1010 => [1,1,1,1] => [1,1,1,1] => [3,1] => 0

1011 => [1,1,2] => [2,1,1] => [2,2] => 1

1100 => [2,2] => [2,2] => [4] => 1

1101 => [2,1,1] => [2,1,1] => [2,2] => 1

1110 => [3,1] => [3,1] => [2,1,1] => 0

1111 => [4] => [4] => [1,1,1,1] => 0

00000 => [5] => [5] => [1,1,1,1,1] => 1

00001 => [4,1] => [4,1] => [2,1,1,1] => 0

00010 => [3,1,1] => [3,1,1] => [4,1] => 0

00011 => [3,2] => [3,2] => [5] => 1

00100 => [2,1,2] => [2,2,1] => [2,2,1] => 1

00101 => [2,1,1,1] => [2,1,1,1] => [3,1,1] => 0

00110 => [2,2,1] => [2,2,1] => [2,2,1] => 1

00111 => [2,3] => [3,2] => [5] => 1

01000 => [1,1,3] => [3,1,1] => [4,1] => 0

01001 => [1,1,2,1] => [2,1,1,1] => [3,1,1] => 0

01010 => [1,1,1,1,1] => [1,1,1,1,1] => [3,2] => 1

01011 => [1,1,1,2] => [2,1,1,1] => [3,1,1] => 0

01100 => [1,2,2] => [2,2,1] => [2,2,1] => 1

01101 => [1,2,1,1] => [2,1,1,1] => [3,1,1] => 0

01110 => [1,3,1] => [3,1,1] => [4,1] => 0

01111 => [1,4] => [4,1] => [2,1,1,1] => 0

10000 => [1,4] => [4,1] => [2,1,1,1] => 0

10001 => [1,3,1] => [3,1,1] => [4,1] => 0

10010 => [1,2,1,1] => [2,1,1,1] => [3,1,1] => 0

10011 => [1,2,2] => [2,2,1] => [2,2,1] => 1

10100 => [1,1,1,2] => [2,1,1,1] => [3,1,1] => 0

10101 => [1,1,1,1,1] => [1,1,1,1,1] => [3,2] => 1

10110 => [1,1,2,1] => [2,1,1,1] => [3,1,1] => 0

10111 => [1,1,3] => [3,1,1] => [4,1] => 0

11000 => [2,3] => [3,2] => [5] => 1

11001 => [2,2,1] => [2,2,1] => [2,2,1] => 1

11010 => [2,1,1,1] => [2,1,1,1] => [3,1,1] => 0

11011 => [2,1,2] => [2,2,1] => [2,2,1] => 1

11100 => [3,2] => [3,2] => [5] => 1

11101 => [3,1,1] => [3,1,1] => [4,1] => 0

11110 => [4,1] => [4,1] => [2,1,1,1] => 0

11111 => [5] => [5] => [1,1,1,1,1] => 1

000000 => [6] => [6] => [1,1,1,1,1,1] => 0

000001 => [5,1] => [5,1] => [2,1,1,1,1] => 1

000010 => [4,1,1] => [4,1,1] => [3,1,1,1] => 1

000011 => [4,2] => [4,2] => [2,2,1,1] => 0

000100 => [3,1,2] => [3,2,1] => [5,1] => 0

000101 => [3,1,1,1] => [3,1,1,1] => [3,3] => 0

000110 => [3,2,1] => [3,2,1] => [5,1] => 0

000111 => [3,3] => [3,3] => [6] => 1

001000 => [2,1,3] => [3,2,1] => [5,1] => 0

001001 => [2,1,2,1] => [2,2,1,1] => [4,1,1] => 0

001010 => [2,1,1,1,1] => [2,1,1,1,1] => [4,2] => 2

001011 => [2,1,1,2] => [2,2,1,1] => [4,1,1] => 0

001100 => [2,2,2] => [2,2,2] => [2,2,2] => 2

001101 => [2,2,1,1] => [2,2,1,1] => [4,1,1] => 0

001110 => [2,3,1] => [3,2,1] => [5,1] => 0

001111 => [2,4] => [4,2] => [2,2,1,1] => 0

010000 => [1,1,4] => [4,1,1] => [3,1,1,1] => 1

010001 => [1,1,3,1] => [3,1,1,1] => [3,3] => 0

010010 => [1,1,2,1,1] => [2,1,1,1,1] => [4,2] => 2

010011 => [1,1,2,2] => [2,2,1,1] => [4,1,1] => 0

010100 => [1,1,1,1,2] => [2,1,1,1,1] => [4,2] => 2

010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [3,2,1] => 1

010110 => [1,1,1,2,1] => [2,1,1,1,1] => [4,2] => 2

010111 => [1,1,1,3] => [3,1,1,1] => [3,3] => 0

011000 => [1,2,3] => [3,2,1] => [5,1] => 0

011001 => [1,2,2,1] => [2,2,1,1] => [4,1,1] => 0

011010 => [1,2,1,1,1] => [2,1,1,1,1] => [4,2] => 2

011011 => [1,2,1,2] => [2,2,1,1] => [4,1,1] => 0

011100 => [1,3,2] => [3,2,1] => [5,1] => 0

011101 => [1,3,1,1] => [3,1,1,1] => [3,3] => 0

011110 => [1,4,1] => [4,1,1] => [3,1,1,1] => 1

011111 => [1,5] => [5,1] => [2,1,1,1,1] => 1

100000 => [1,5] => [5,1] => [2,1,1,1,1] => 1

100001 => [1,4,1] => [4,1,1] => [3,1,1,1] => 1

100010 => [1,3,1,1] => [3,1,1,1] => [3,3] => 0

100011 => [1,3,2] => [3,2,1] => [5,1] => 0

100100 => [1,2,1,2] => [2,2,1,1] => [4,1,1] => 0

100101 => [1,2,1,1,1] => [2,1,1,1,1] => [4,2] => 2

100110 => [1,2,2,1] => [2,2,1,1] => [4,1,1] => 0

100111 => [1,2,3] => [3,2,1] => [5,1] => 0

101000 => [1,1,1,3] => [3,1,1,1] => [3,3] => 0

101001 => [1,1,1,2,1] => [2,1,1,1,1] => [4,2] => 2

101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [3,2,1] => 1

101011 => [1,1,1,1,2] => [2,1,1,1,1] => [4,2] => 2

101100 => [1,1,2,2] => [2,2,1,1] => [4,1,1] => 0

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Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Map

Loehr-Warrington

Description

Return a partition whose diagonal inversion number is the length of the preimage.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.