- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤ

Values

0 => [2] => 1

1 => [1,1] => 1

00 => [3] => 1

01 => [2,1] => 2

10 => [1,2] => 2

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

000 => [4] => 1

001 => [3,1] => 3

010 => [2,2] => 5

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

100 => [1,3] => 3

101 => [1,2,1] => 5

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

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

0000 => [5] => 1

0001 => [4,1] => 4

0010 => [3,2] => 9

0011 => [3,1,1] => 6

0100 => [2,3] => 9

0101 => [2,2,1] => 16

0110 => [2,1,2] => 11

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

1000 => [1,4] => 4

1001 => [1,3,1] => 11

1010 => [1,2,2] => 16

1011 => [1,2,1,1] => 9

1100 => [1,1,3] => 6

1101 => [1,1,2,1] => 9

1110 => [1,1,1,2] => 4

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

00000 => [6] => 1

00001 => [5,1] => 5

00010 => [4,2] => 14

00011 => [4,1,1] => 10

00100 => [3,3] => 19

00101 => [3,2,1] => 35

00110 => [3,1,2] => 26

00111 => [3,1,1,1] => 10

01000 => [2,4] => 14

01001 => [2,3,1] => 40

01010 => [2,2,2] => 61

01011 => [2,2,1,1] => 35

01100 => [2,1,3] => 26

01101 => [2,1,2,1] => 40

01110 => [2,1,1,2] => 19

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

10000 => [1,5] => 5

10001 => [1,4,1] => 19

10010 => [1,3,2] => 40

10011 => [1,3,1,1] => 26

10100 => [1,2,3] => 35

10101 => [1,2,2,1] => 61

10110 => [1,2,1,2] => 40

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

11000 => [1,1,4] => 10

11001 => [1,1,3,1] => 26

11010 => [1,1,2,2] => 35

11011 => [1,1,2,1,1] => 19

11100 => [1,1,1,3] => 10

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

11110 => [1,1,1,1,2] => 5

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

000000 => [7] => 1

000001 => [6,1] => 6

000010 => [5,2] => 20

000011 => [5,1,1] => 15

000100 => [4,3] => 34

000101 => [4,2,1] => 64

000110 => [4,1,2] => 50

000111 => [4,1,1,1] => 20

001000 => [3,4] => 34

001001 => [3,3,1] => 99

001010 => [3,2,2] => 155

001011 => [3,2,1,1] => 90

001100 => [3,1,3] => 71

001101 => [3,1,2,1] => 111

001110 => [3,1,1,2] => 55

001111 => [3,1,1,1,1] => 15

010000 => [2,5] => 20

010001 => [2,4,1] => 78

010010 => [2,3,2] => 169

010011 => [2,3,1,1] => 111

010100 => [2,2,3] => 155

010101 => [2,2,2,1] => 272

010110 => [2,2,1,2] => 181

010111 => [2,2,1,1,1] => 64

011000 => [2,1,4] => 50

011001 => [2,1,3,1] => 132

011010 => [2,1,2,2] => 181

011011 => [2,1,2,1,1] => 99

011100 => [2,1,1,3] => 55

011101 => [2,1,1,2,1] => 78

011110 => [2,1,1,1,2] => 29

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

100000 => [1,6] => 6

100001 => [1,5,1] => 29

100010 => [1,4,2] => 78

100011 => [1,4,1,1] => 55

100100 => [1,3,3] => 99

100101 => [1,3,2,1] => 181

100110 => [1,3,1,2] => 132

>>> Load all 633 entries. <<<

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Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 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.