- FindStat

Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤ

Values

{{1}} => [1] => [1,0] => 10 => 1

{{1,2}} => [2] => [1,0,1,0] => 1010 => 1

{{1},{2}} => [1,1] => [1,1,0,0] => 1100 => 2

{{1,2,3}} => [3] => [1,0,1,0,1,0] => 101010 => 1

{{1,2},{3}} => [2,1] => [1,0,1,1,0,0] => 101100 => 2

{{1,3},{2}} => [2,1] => [1,0,1,1,0,0] => 101100 => 2

{{1},{2,3}} => [2,1] => [1,0,1,1,0,0] => 101100 => 2

{{1},{2},{3}} => [1,1,1] => [1,1,0,1,0,0] => 110100 => 2

{{1,2,3,4}} => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 1

{{1,2,3},{4}} => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 2

{{1,2,4},{3}} => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 2

{{1,2},{3,4}} => [2,2] => [1,1,1,0,0,0] => 111000 => 3

{{1,2},{3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1,3,4},{2}} => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 2

{{1,3},{2,4}} => [2,2] => [1,1,1,0,0,0] => 111000 => 3

{{1,3},{2},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1,4},{2,3}} => [2,2] => [1,1,1,0,0,0] => 111000 => 3

{{1},{2,3,4}} => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 2

{{1},{2,3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1,4},{2},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1},{2,4},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1},{2},{3,4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 2

{{1},{2},{3},{4}} => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 2

{{1,2,3,4,5}} => [5] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1

{{1,2,3,4},{5}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

{{1,2,3,5},{4}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

{{1,2,3},{4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,2,3},{4},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,2,4,5},{3}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

{{1,2,4},{3,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,2,4},{3},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,2,5},{3,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,2},{3,4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,2},{3,4},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,2,5},{3},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,2},{3,5},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,2},{3},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,3,4,5},{2}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

{{1,3,4},{2,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,3,4},{2},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,3,5},{2,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,3},{2,4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,3},{2,4},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,3,5},{2},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,3},{2,5},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,3},{2},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,4,5},{2,3}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,4},{2,3,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1,4},{2,3},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,5},{2,3,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 3

{{1},{2,3,4,5}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

{{1},{2,3,4},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,5},{2,3},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2,3,5},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1},{2,3},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2,3},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,4,5},{2},{3}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1,4},{2,5},{3}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,4},{2},{3,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,5},{2,4},{3}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2,4,5},{3}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1},{2,4},{3,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2,4},{3},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,5},{2},{3,4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2,5},{3,4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 3

{{1},{2},{3,4,5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

{{1},{2},{3,4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1},{2,5},{3},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1},{2},{3,5},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1},{2},{3},{4,5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 2

{{1,2,3,4,5,6}} => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 1

{{1,2,3,4,5},{6}} => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 2

{{1,2,3,4,6},{5}} => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 2

{{1,2,3,4},{5,6}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

{{1,2,3,4},{5},{6}} => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

{{1,2,3,5,6},{4}} => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 2

{{1,2,3,5},{4,6}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

{{1,2,3,5},{4},{6}} => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

{{1,2,3,6},{4,5}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

{{1,2,3},{4,5,6}} => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 3

{{1,2,3},{4,5},{6}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,3,6},{4},{5}} => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

{{1,2,3},{4,6},{5}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,3},{4},{5,6}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 2

{{1,2,4,5,6},{3}} => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 2

{{1,2,4,5},{3,6}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

{{1,2,4,5},{3},{6}} => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

{{1,2,4,6},{3,5}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

{{1,2,4},{3,5,6}} => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 3

{{1,2,4},{3,5},{6}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,4,6},{3},{5}} => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

{{1,2,4},{3,6},{5}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,4},{3},{5,6}} => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 2

{{1,2,5,6},{3,4}} => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

>>> Load all 2494 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

$F_{1} = q$

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

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

$F_{4} = q + 11\ q^{2} + 3\ q^{3}$

$F_{5} = q + 26\ q^{2} + 25\ q^{3}$

$F_{6} = q + 57\ q^{2} + 130\ q^{3} + 15\ q^{4}$

Description

The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to binary word

Description

Return the Dyck word as binary word.