- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤ

Values

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

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

00 => [3] => [3] => [3] => 0

01 => [2,1] => [2,1] => [1,1,1] => 1

10 => [1,2] => [2,1] => [1,1,1] => 1

11 => [1,1,1] => [1,1,1] => [2,1] => 0

000 => [4] => [4] => [2,2] => 1

001 => [3,1] => [3,1] => [3,1] => 0

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

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

100 => [1,3] => [3,1] => [3,1] => 0

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

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

111 => [1,1,1,1] => [1,1,1,1] => [4] => 0

0000 => [5] => [5] => [5] => 0

0001 => [4,1] => [4,1] => [2,2,1] => 1

0010 => [3,2] => [3,2] => [3,1,1] => 1

0011 => [3,1,1] => [3,1,1] => [3,2] => 0

0100 => [2,3] => [3,2] => [3,1,1] => 1

0101 => [2,2,1] => [2,2,1] => [1,1,1,1,1] => 2

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

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

1000 => [1,4] => [4,1] => [2,2,1] => 1

1001 => [1,3,1] => [3,1,1] => [3,2] => 0

1010 => [1,2,2] => [2,2,1] => [1,1,1,1,1] => 2

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

1100 => [1,1,3] => [3,1,1] => [3,2] => 0

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

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

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

00000 => [6] => [6] => [3,3] => 1

00001 => [5,1] => [5,1] => [5,1] => 0

00010 => [4,2] => [4,2] => [2,2,1,1] => 2

00011 => [4,1,1] => [4,1,1] => [2,2,2] => 1

00100 => [3,3] => [3,3] => [6] => 0

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

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

00111 => [3,1,1,1] => [3,1,1,1] => [3,2,1] => 0

01000 => [2,4] => [4,2] => [2,2,1,1] => 2

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

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

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

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

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

01110 => [2,1,1,2] => [2,2,1,1] => [2,1,1,1,1] => 2

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

10000 => [1,5] => [5,1] => [5,1] => 0

10001 => [1,4,1] => [4,1,1] => [2,2,2] => 1

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

10011 => [1,3,1,1] => [3,1,1,1] => [3,2,1] => 0

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

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

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

10111 => [1,2,1,1,1] => [2,1,1,1,1] => [4,1,1] => 1

11000 => [1,1,4] => [4,1,1] => [2,2,2] => 1

11001 => [1,1,3,1] => [3,1,1,1] => [3,2,1] => 0

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

11011 => [1,1,2,1,1] => [2,1,1,1,1] => [4,1,1] => 1

11100 => [1,1,1,3] => [3,1,1,1] => [3,2,1] => 0

11101 => [1,1,1,2,1] => [2,1,1,1,1] => [4,1,1] => 1

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

11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [4,2] => 0

000000 => [7] => [7] => [7] => 0

000001 => [6,1] => [6,1] => [3,3,1] => 1

000010 => [5,2] => [5,2] => [5,1,1] => 1

000011 => [5,1,1] => [5,1,1] => [5,2] => 0

000100 => [4,3] => [4,3] => [3,2,2] => 1

000101 => [4,2,1] => [4,2,1] => [2,2,1,1,1] => 2

000110 => [4,1,2] => [4,2,1] => [2,2,1,1,1] => 2

000111 => [4,1,1,1] => [4,1,1,1] => [2,2,2,1] => 1

001000 => [3,4] => [4,3] => [3,2,2] => 1

001001 => [3,3,1] => [3,3,1] => [6,1] => 0

001010 => [3,2,2] => [3,2,2] => [3,1,1,1,1] => 2

001011 => [3,2,1,1] => [3,2,1,1] => [3,2,1,1] => 1

001100 => [3,1,3] => [3,3,1] => [6,1] => 0

001101 => [3,1,2,1] => [3,2,1,1] => [3,2,1,1] => 1

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

001111 => [3,1,1,1,1] => [3,1,1,1,1] => [4,3] => 0

010000 => [2,5] => [5,2] => [5,1,1] => 1

010001 => [2,4,1] => [4,2,1] => [2,2,1,1,1] => 2

010010 => [2,3,2] => [3,2,2] => [3,1,1,1,1] => 2

010011 => [2,3,1,1] => [3,2,1,1] => [3,2,1,1] => 1

010100 => [2,2,3] => [3,2,2] => [3,1,1,1,1] => 2

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

010110 => [2,2,1,2] => [2,2,2,1] => [1,1,1,1,1,1,1] => 3

010111 => [2,2,1,1,1] => [2,2,1,1,1] => [2,1,1,1,1,1] => 2

011000 => [2,1,4] => [4,2,1] => [2,2,1,1,1] => 2

011001 => [2,1,3,1] => [3,2,1,1] => [3,2,1,1] => 1

011010 => [2,1,2,2] => [2,2,2,1] => [1,1,1,1,1,1,1] => 3

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

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

011101 => [2,1,1,2,1] => [2,2,1,1,1] => [2,1,1,1,1,1] => 2

011110 => [2,1,1,1,2] => [2,2,1,1,1] => [2,1,1,1,1,1] => 2

011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1] => [4,1,1,1] => 1

100000 => [1,6] => [6,1] => [3,3,1] => 1

100001 => [1,5,1] => [5,1,1] => [5,2] => 0

100010 => [1,4,2] => [4,2,1] => [2,2,1,1,1] => 2

100011 => [1,4,1,1] => [4,1,1,1] => [2,2,2,1] => 1

100100 => [1,3,3] => [3,3,1] => [6,1] => 0

100101 => [1,3,2,1] => [3,2,1,1] => [3,2,1,1] => 1

100110 => [1,3,1,2] => [3,2,1,1] => [3,2,1,1] => 1

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Description

The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with St000143The largest repeated part of a partition. and St000149The number of cells of the partition whose leg is zero and arm is odd., see [1].

Map

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

Map

to partition

Description

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

Map

Glaisher-Franklin

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].