- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001658: Skew partitions ⟶ ℤ

Values

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

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

00 => [3] => [[3],[]] => 4

01 => [2,1] => [[2,2],[1]] => 5

10 => [1,2] => [[2,1],[]] => 5

11 => [1,1,1] => [[1,1,1],[]] => 4

000 => [4] => [[4],[]] => 5

001 => [3,1] => [[3,3],[2]] => 7

010 => [2,2] => [[3,2],[1]] => 8

011 => [2,1,1] => [[2,2,2],[1,1]] => 7

100 => [1,3] => [[3,1],[]] => 7

101 => [1,2,1] => [[2,2,1],[1]] => 8

110 => [1,1,2] => [[2,1,1],[]] => 7

111 => [1,1,1,1] => [[1,1,1,1],[]] => 5

0000 => [5] => [[5],[]] => 6

0001 => [4,1] => [[4,4],[3]] => 9

0010 => [3,2] => [[4,3],[2]] => 11

0011 => [3,1,1] => [[3,3,3],[2,2]] => 10

0100 => [2,3] => [[4,2],[1]] => 11

0101 => [2,2,1] => [[3,3,2],[2,1]] => 13

0110 => [2,1,2] => [[3,2,2],[1,1]] => 12

0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 9

1000 => [1,4] => [[4,1],[]] => 9

1001 => [1,3,1] => [[3,3,1],[2]] => 12

1010 => [1,2,2] => [[3,2,1],[1]] => 13

1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 11

1100 => [1,1,3] => [[3,1,1],[]] => 10

1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 11

1110 => [1,1,1,2] => [[2,1,1,1],[]] => 9

1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 6

00000 => [6] => [[6],[]] => 7

00001 => [5,1] => [[5,5],[4]] => 11

00010 => [4,2] => [[5,4],[3]] => 14

00011 => [4,1,1] => [[4,4,4],[3,3]] => 13

00100 => [3,3] => [[5,3],[2]] => 15

00101 => [3,2,1] => [[4,4,3],[3,2]] => 18

00110 => [3,1,2] => [[4,3,3],[2,2]] => 17

00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => 13

01000 => [2,4] => [[5,2],[1]] => 14

01001 => [2,3,1] => [[4,4,2],[3,1]] => 19

01010 => [2,2,2] => [[4,3,2],[2,1]] => 21

01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => 18

01100 => [2,1,3] => [[4,2,2],[1,1]] => 17

01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => 19

01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => 16

01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => 11

10000 => [1,5] => [[5,1],[]] => 11

10001 => [1,4,1] => [[4,4,1],[3]] => 16

10010 => [1,3,2] => [[4,3,1],[2]] => 19

10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 17

10100 => [1,2,3] => [[4,2,1],[1]] => 18

10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 21

10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 19

10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 14

11000 => [1,1,4] => [[4,1,1],[]] => 13

11001 => [1,1,3,1] => [[3,3,1,1],[2]] => 17

11010 => [1,1,2,2] => [[3,2,1,1],[1]] => 18

11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => 15

11100 => [1,1,1,3] => [[3,1,1,1],[]] => 13

11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => 14

11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => 11

11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 7

000000 => [7] => [[7],[]] => 8

000001 => [6,1] => [[6,6],[5]] => 13

000010 => [5,2] => [[6,5],[4]] => 17

000011 => [5,1,1] => [[5,5,5],[4,4]] => 16

000100 => [4,3] => [[6,4],[3]] => 19

000101 => [4,2,1] => [[5,5,4],[4,3]] => 23

000110 => [4,1,2] => [[5,4,4],[3,3]] => 22

000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => 17

001000 => [3,4] => [[6,3],[2]] => 19

001001 => [3,3,1] => [[5,5,3],[4,2]] => 26

001010 => [3,2,2] => [[5,4,3],[3,2]] => 29

001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => 25

001100 => [3,1,3] => [[5,3,3],[2,2]] => 24

001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => 27

001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => 23

001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => 16

010000 => [2,5] => [[6,2],[1]] => 17

010001 => [2,4,1] => [[5,5,2],[4,1]] => 25

010010 => [2,3,2] => [[5,4,2],[3,1]] => 30

010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => 27

010100 => [2,2,3] => [[5,3,2],[2,1]] => 29

010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => 34

010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => 31

010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => 23

011000 => [2,1,4] => [[5,2,2],[1,1]] => 22

011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => 29

011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => 31

011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => 26

011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => 23

011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => 25

011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => 20

011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => 13

100000 => [1,6] => [[6,1],[]] => 13

100001 => [1,5,1] => [[5,5,1],[4]] => 20

100010 => [1,4,2] => [[5,4,1],[3]] => 25

100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => 23

100100 => [1,3,3] => [[5,3,1],[2]] => 26

100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => 31

100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => 29

>>> Load all 127 entries. <<<

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$F_{1} = 2\ q^{3}$

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

$F_{3} = 2\ q^{5} + 4\ q^{7} + 2\ q^{8}$

$F_{4} = 2\ q^{6} + 4\ q^{9} + 2\ q^{10} + 4\ q^{11} + 2\ q^{12} + 2\ q^{13}$

$F_{5} = 2\ q^{7} + 4\ q^{11} + 4\ q^{13} + 4\ q^{14} + 2\ q^{15} + 2\ q^{16} + 4\ q^{17} + 4\ q^{18} + 4\ q^{19} + 2\ q^{21}$

$F_{6} = 2\ q^{8} + 4\ q^{13} + 4\ q^{16} + 6\ q^{17} + 4\ q^{19} + 2\ q^{20} + 4\ q^{22} + 8\ q^{23} + 2\ q^{24} + 6\ q^{25} + 4\ q^{26} + 4\ q^{27} + 6\ q^{29} + 2\ q^{30} + 4\ q^{31} + 2\ q^{34}$

Description

The total number of rook placements on a Ferrers board.

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 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.