- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001035: Dyck paths ⟶ ℤ

Values

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

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

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

01 => [2,1] => [2,1] => [1,0,1,1,0,0] => 1

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

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

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

001 => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 1

010 => [2,2] => [2,2] => [1,1,1,0,0,0] => 1

011 => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1

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

101 => [1,2,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1

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

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

0000 => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0

0001 => [4,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1

0010 => [3,2] => [3,2] => [1,0,1,1,1,0,0,0] => 1

0011 => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1

0100 => [2,3] => [3,2] => [1,0,1,1,1,0,0,0] => 1

0101 => [2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1

0110 => [2,1,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1

0111 => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1

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

1001 => [1,3,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1

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

1011 => [1,2,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1

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

1101 => [1,1,2,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1

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

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

00000 => [6] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0

00001 => [5,1] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1

00010 => [4,2] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1

00011 => [4,1,1] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1

00100 => [3,3] => [3,3] => [1,1,1,0,1,0,0,0] => 1

00101 => [3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

00110 => [3,1,2] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

00111 => [3,1,1,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1

01000 => [2,4] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1

01001 => [2,3,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

01010 => [2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => 1

01011 => [2,2,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

01100 => [2,1,3] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

01101 => [2,1,2,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

01110 => [2,1,1,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

01111 => [2,1,1,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1

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

10001 => [1,4,1] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1

10010 => [1,3,2] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

10011 => [1,3,1,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1

10100 => [1,2,3] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1

10101 => [1,2,2,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

10110 => [1,2,1,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

10111 => [1,2,1,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1

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

11001 => [1,1,3,1] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1

11010 => [1,1,2,2] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1

11011 => [1,1,2,1,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1

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

11101 => [1,1,1,2,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1

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

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

000000 => [7] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0

000001 => [6,1] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1

000010 => [5,2] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1

000011 => [5,1,1] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1

000100 => [4,3] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

000101 => [4,2,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1

000110 => [4,1,2] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1

000111 => [4,1,1,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1

001000 => [3,4] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1

001001 => [3,3,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

001010 => [3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1

001011 => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

001100 => [3,1,3] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

001101 => [3,1,2,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

001110 => [3,1,1,2] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

001111 => [3,1,1,1,1] => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1

010000 => [2,5] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1

010001 => [2,4,1] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1

010010 => [2,3,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1

010011 => [2,3,1,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

010100 => [2,2,3] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1

010101 => [2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1

010110 => [2,2,1,2] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1

010111 => [2,2,1,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1

011000 => [2,1,4] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1

011001 => [2,1,3,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

011010 => [2,1,2,2] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1

011011 => [2,1,2,1,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1

011100 => [2,1,1,3] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

011101 => [2,1,1,2,1] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1

011110 => [2,1,1,1,2] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1

011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 1

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

100001 => [1,5,1] => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1

100010 => [1,4,2] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1

100011 => [1,4,1,1] => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1

100100 => [1,3,3] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1

100101 => [1,3,2,1] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

100110 => [1,3,1,2] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1

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Description

The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].

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 partition

Description

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

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.