- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤ

Values

0 => [1] => [1] => 1

1 => [1] => [1] => 1

00 => [2] => [2] => 2

01 => [1,1] => [1,1] => 1

10 => [1,1] => [1,1] => 1

11 => [2] => [2] => 2

000 => [3] => [3] => 2

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

010 => [1,1,1] => [1,1,1] => 1

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

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

101 => [1,1,1] => [1,1,1] => 1

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

111 => [3] => [3] => 2

0000 => [4] => [4] => 2

0001 => [3,1] => [3,1] => 4

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

0011 => [2,2] => [2,2] => 3

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

0101 => [1,1,1,1] => [1,1,1,1] => 1

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

0111 => [1,3] => [3,1] => 4

1000 => [1,3] => [3,1] => 4

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

1010 => [1,1,1,1] => [1,1,1,1] => 1

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

1100 => [2,2] => [2,2] => 3

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

1110 => [3,1] => [3,1] => 4

1111 => [4] => [4] => 2

00000 => [5] => [5] => 2

00001 => [4,1] => [4,1] => 3

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

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

00100 => [2,1,2] => [2,2,1] => 3

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

00110 => [2,2,1] => [2,2,1] => 3

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

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

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

01010 => [1,1,1,1,1] => [1,1,1,1,1] => 1

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

01100 => [1,2,2] => [2,2,1] => 3

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

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

01111 => [1,4] => [4,1] => 3

10000 => [1,4] => [4,1] => 3

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

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

10011 => [1,2,2] => [2,2,1] => 3

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

10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1

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

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

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

11001 => [2,2,1] => [2,2,1] => 3

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

11011 => [2,1,2] => [2,2,1] => 3

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

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

11110 => [4,1] => [4,1] => 3

11111 => [5] => [5] => 2

000000 => [6] => [6] => 2

000001 => [5,1] => [5,1] => 3

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

000011 => [4,2] => [4,2] => 5

000100 => [3,1,2] => [3,2,1] => 6

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

000110 => [3,2,1] => [3,2,1] => 6

000111 => [3,3] => [3,3] => 4

001000 => [2,1,3] => [3,2,1] => 6

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

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

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

001100 => [2,2,2] => [2,2,2] => 3

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

001110 => [2,3,1] => [3,2,1] => 6

001111 => [2,4] => [4,2] => 5

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

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

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

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

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

010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

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

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

011000 => [1,2,3] => [3,2,1] => 6

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

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

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

011100 => [1,3,2] => [3,2,1] => 6

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

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

011111 => [1,5] => [5,1] => 3

100000 => [1,5] => [5,1] => 3

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

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

100011 => [1,3,2] => [3,2,1] => 6

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

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

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

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Description

The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].

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 $w$.