- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000869: Integer partitions ⟶ ℤ

Values

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

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

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

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

10 => [1,2] => [[2,1],[]] => [] => 0

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

000 => [4] => [[4],[]] => [] => 0

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

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

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

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

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

110 => [1,1,2] => [[2,1,1],[]] => [] => 0

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

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

0001 => [4,1] => [[4,4],[3]] => [3] => 6

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

0011 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 8

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

0101 => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 5

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

0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 6

1000 => [1,4] => [[4,1],[]] => [] => 0

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

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

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

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

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

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

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

00000 => [6] => [[6],[]] => [] => 0

00001 => [5,1] => [[5,5],[4]] => [4] => 10

00010 => [4,2] => [[5,4],[3]] => [3] => 6

00011 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 15

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

00101 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 11

00110 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 8

00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 15

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

01001 => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 8

01010 => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 5

01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 11

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

01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 8

01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 6

01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 10

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

10001 => [1,4,1] => [[4,4,1],[3]] => [3] => 6

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

10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 8

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

10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 5

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

10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 6

11000 => [1,1,4] => [[4,1,1],[]] => [] => 0

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

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

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

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

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

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

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

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

000001 => [6,1] => [[6,6],[5]] => [5] => 15

000010 => [5,2] => [[6,5],[4]] => [4] => 10

000011 => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 24

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

000101 => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 19

000110 => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 15

000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 27

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

001001 => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 15

001010 => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 11

001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 22

001100 => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 8

001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 18

001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 15

001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 24

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

010001 => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 12

010010 => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 8

010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 18

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

010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 14

010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 11

010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 19

011000 => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 3

011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 11

011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 8

011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 15

011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 6

011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 12

011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 10

011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 15

100000 => [1,6] => [[6,1],[]] => [] => 0

100001 => [1,5,1] => [[5,5,1],[4]] => [4] => 10

100010 => [1,4,2] => [[5,4,1],[3]] => [3] => 6

100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 15

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

100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 11

100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 8

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$F_{1} = 2$

$F_{2} = 3 + q$

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

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

$F_{5} = 6 + 4\ q + 6\ q^{3} + 2\ q^{5} + 4\ q^{6} + 4\ q^{8} + 2\ q^{10} + 2\ q^{11} + 2\ q^{15}$

$F_{6} = 7 + 5\ q + 8\ q^{3} + 3\ q^{5} + 6\ q^{6} + 7\ q^{8} + 4\ q^{10} + 5\ q^{11} + 2\ q^{12} + q^{14} + 8\ q^{15} + 2\ q^{18} + 2\ q^{19} + q^{22} + 2\ q^{24} + q^{27}$

Description

The sum of the hook lengths of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the sum of all hook lengths of a 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

inner shape

Description

The inner shape of a 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.