- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤ

Values

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

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

00 => [3] => [3] => [[1,2,3]] => 0

01 => [2,1] => [2,1] => [[1,3],[2]] => 1

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

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

000 => [4] => [4] => [[1,2,3,4]] => 0

001 => [3,1] => [3,1] => [[1,3,4],[2]] => 1

010 => [2,2] => [2,2] => [[1,2],[3,4]] => 1

011 => [2,1,1] => [2,1,1] => [[1,4],[2],[3]] => 2

100 => [1,3] => [3,1] => [[1,3,4],[2]] => 1

101 => [1,2,1] => [2,1,1] => [[1,4],[2],[3]] => 2

110 => [1,1,2] => [2,1,1] => [[1,4],[2],[3]] => 2

111 => [1,1,1,1] => [1,1,1,1] => [[1],[2],[3],[4]] => 3

0000 => [5] => [5] => [[1,2,3,4,5]] => 0

0001 => [4,1] => [4,1] => [[1,3,4,5],[2]] => 1

0010 => [3,2] => [3,2] => [[1,2,5],[3,4]] => 1

0011 => [3,1,1] => [3,1,1] => [[1,4,5],[2],[3]] => 2

0100 => [2,3] => [3,2] => [[1,2,5],[3,4]] => 1

0101 => [2,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => 2

0110 => [2,1,2] => [2,2,1] => [[1,3],[2,5],[4]] => 2

0111 => [2,1,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 3

1000 => [1,4] => [4,1] => [[1,3,4,5],[2]] => 1

1001 => [1,3,1] => [3,1,1] => [[1,4,5],[2],[3]] => 2

1010 => [1,2,2] => [2,2,1] => [[1,3],[2,5],[4]] => 2

1011 => [1,2,1,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 3

1100 => [1,1,3] => [3,1,1] => [[1,4,5],[2],[3]] => 2

1101 => [1,1,2,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 3

1110 => [1,1,1,2] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 3

1111 => [1,1,1,1,1] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 4

00000 => [6] => [6] => [[1,2,3,4,5,6]] => 0

00001 => [5,1] => [5,1] => [[1,3,4,5,6],[2]] => 1

00010 => [4,2] => [4,2] => [[1,2,5,6],[3,4]] => 1

00011 => [4,1,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

00100 => [3,3] => [3,3] => [[1,2,3],[4,5,6]] => 1

00101 => [3,2,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

00110 => [3,1,2] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

00111 => [3,1,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

01000 => [2,4] => [4,2] => [[1,2,5,6],[3,4]] => 1

01001 => [2,3,1] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

01010 => [2,2,2] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2

01011 => [2,2,1,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

01100 => [2,1,3] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

01101 => [2,1,2,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

01110 => [2,1,1,2] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

01111 => [2,1,1,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

10000 => [1,5] => [5,1] => [[1,3,4,5,6],[2]] => 1

10001 => [1,4,1] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

10010 => [1,3,2] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

10011 => [1,3,1,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

10100 => [1,2,3] => [3,2,1] => [[1,3,6],[2,5],[4]] => 2

10101 => [1,2,2,1] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

10110 => [1,2,1,2] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

10111 => [1,2,1,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

11000 => [1,1,4] => [4,1,1] => [[1,4,5,6],[2],[3]] => 2

11001 => [1,1,3,1] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

11010 => [1,1,2,2] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 3

11011 => [1,1,2,1,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

11100 => [1,1,1,3] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 3

11101 => [1,1,1,2,1] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

11110 => [1,1,1,1,2] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 4

11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => 5

000000 => [7] => [7] => [[1,2,3,4,5,6,7]] => 0

000001 => [6,1] => [6,1] => [[1,3,4,5,6,7],[2]] => 1

000010 => [5,2] => [5,2] => [[1,2,5,6,7],[3,4]] => 1

000011 => [5,1,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 2

000100 => [4,3] => [4,3] => [[1,2,3,7],[4,5,6]] => 1

000101 => [4,2,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 2

000110 => [4,1,2] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 2

000111 => [4,1,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 3

001000 => [3,4] => [4,3] => [[1,2,3,7],[4,5,6]] => 1

001001 => [3,3,1] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 2

001010 => [3,2,2] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 2

001011 => [3,2,1,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

001100 => [3,1,3] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 2

001101 => [3,1,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

001110 => [3,1,1,2] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

001111 => [3,1,1,1,1] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 4

010000 => [2,5] => [5,2] => [[1,2,5,6,7],[3,4]] => 1

010001 => [2,4,1] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 2

010010 => [2,3,2] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 2

010011 => [2,3,1,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

010100 => [2,2,3] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 2

010101 => [2,2,2,1] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 3

010110 => [2,2,1,2] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 3

010111 => [2,2,1,1,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 4

011000 => [2,1,4] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 2

011001 => [2,1,3,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

011010 => [2,1,2,2] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 3

011011 => [2,1,2,1,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 4

011100 => [2,1,1,3] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

011101 => [2,1,1,2,1] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 4

011110 => [2,1,1,1,2] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 4

011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => 5

100000 => [1,6] => [6,1] => [[1,3,4,5,6,7],[2]] => 1

100001 => [1,5,1] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 2

100010 => [1,4,2] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 2

100011 => [1,4,1,1] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 3

100100 => [1,3,3] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 2

100101 => [1,3,2,1] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

100110 => [1,3,1,2] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 3

>>> Load all 1136 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

Description

The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.

Map

to partition

Description

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

Map

reading tableau

Description

Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a 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.