- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤ

Values

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

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

00 => [3] => [[3],[]] => [3] => 1

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

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

11 => [1,1,1] => [[1,1,1],[]] => [1,1,1] => 3

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

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

010 => [2,2] => [[3,2],[1]] => [3,2] => 1

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

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

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

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

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

0000 => [5] => [[5],[]] => [5] => 1

0001 => [4,1] => [[4,4],[3]] => [4,4] => 2

0010 => [3,2] => [[4,3],[2]] => [4,3] => 1

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

0100 => [2,3] => [[4,2],[1]] => [4,2] => 1

0101 => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 2

0110 => [2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => 1

0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => 4

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

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

1010 => [1,2,2] => [[3,2,1],[1]] => [3,2,1] => 1

1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => 3

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

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

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

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

00001 => [5,1] => [[5,5],[4]] => [5,5] => 2

00010 => [4,2] => [[5,4],[3]] => [5,4] => 1

00011 => [4,1,1] => [[4,4,4],[3,3]] => [4,4,4] => 3

00100 => [3,3] => [[5,3],[2]] => [5,3] => 1

00101 => [3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => 2

00110 => [3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => 1

00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => 4

01000 => [2,4] => [[5,2],[1]] => [5,2] => 1

01001 => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 2

01010 => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1

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

01100 => [2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => 1

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

01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => 1

01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => 5

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

10001 => [1,4,1] => [[4,4,1],[3]] => [4,4,1] => 2

10010 => [1,3,2] => [[4,3,1],[2]] => [4,3,1] => 1

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

10100 => [1,2,3] => [[4,2,1],[1]] => [4,2,1] => 1

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

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

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

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

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

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

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

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

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

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

11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => 6

000000 => [7] => [[7],[]] => [7] => 1

000001 => [6,1] => [[6,6],[5]] => [6,6] => 2

000010 => [5,2] => [[6,5],[4]] => [6,5] => 1

000100 => [4,3] => [[6,4],[3]] => [6,4] => 1

001000 => [3,4] => [[6,3],[2]] => [6,3] => 1

001010 => [3,2,2] => [[5,4,3],[3,2]] => [5,4,3] => 1

001100 => [3,1,3] => [[5,3,3],[2,2]] => [5,3,3] => 1

010000 => [2,5] => [[6,2],[1]] => [6,2] => 1

010001 => [2,4,1] => [[5,5,2],[4,1]] => [5,5,2] => 2

010010 => [2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => 1

010100 => [2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => 1

010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => 1

011000 => [2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => 1

011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [4,4,2,2] => 2

011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => 1

011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => 1

011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [3,3,2,2,2] => 2

011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [3,2,2,2,2] => 1

011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => 6

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

100001 => [1,5,1] => [[5,5,1],[4]] => [5,5,1] => 2

100010 => [1,4,2] => [[5,4,1],[3]] => [5,4,1] => 1

100100 => [1,3,3] => [[5,3,1],[2]] => [5,3,1] => 1

100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => 2

100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => 1

101000 => [1,2,4] => [[5,2,1],[1]] => [5,2,1] => 1

101001 => [1,2,3,1] => [[4,4,2,1],[3,1]] => [4,4,2,1] => 2

101010 => [1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => 1

101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => 3

101100 => [1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => 1

101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => 2

101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => 1

101111 => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [2,2,2,2,2,1] => 5

110000 => [1,1,5] => [[5,1,1],[]] => [5,1,1] => 1

110001 => [1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => 2

110010 => [1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => 1

110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => 3

110100 => [1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => 1

110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => 2

>>> Load all 161 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q + q^{2}$

$F_{2} = 2\ q + q^{2} + q^{3}$

$F_{3} = 4\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{4} = 8\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{5} = 16\ q + 8\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

Description

The multiplicity of the largest part of an integer 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.

Map

outer shape

Description

The outer shape of the skew partition.

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.