- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤ

Values

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

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

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

11 => [2] => [1,1,0,0] => 0

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

001 => [2,1] => [1,1,0,0,1,0] => 1

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

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

100 => [1,2] => [1,0,1,1,0,0] => 1

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

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

111 => [3] => [1,1,1,0,0,0] => 1

0000 => [4] => [1,1,1,1,0,0,0,0] => 1

0001 => [3,1] => [1,1,1,0,0,0,1,0] => 1

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

0011 => [2,2] => [1,1,0,0,1,1,0,0] => 2

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

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

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

0111 => [1,3] => [1,0,1,1,1,0,0,0] => 1

1000 => [1,3] => [1,0,1,1,1,0,0,0] => 1

1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

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

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

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

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

1110 => [3,1] => [1,1,1,0,0,0,1,0] => 1

1111 => [4] => [1,1,1,1,0,0,0,0] => 1

00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 1

00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1

00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1

00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2

00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

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

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

00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1

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

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

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

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

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

01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2

01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2

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

10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

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

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

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

10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1

11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3

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

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

11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2

11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1

11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1

11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => 1

000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1

000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1

000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1

000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2

000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2

000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1

000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2

001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2

001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3

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

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

001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

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

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

001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2

010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1

010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2

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

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

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

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

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

010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1

011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4

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

011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3

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

011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2

011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2

011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2

100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2

100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3

100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3

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

100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4

100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1

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$F_{2} = 4$

$F_{3} = 2 + 6\ q$

$F_{4} = 2 + 10\ q + 4\ q^{2}$

$F_{5} = 2 + 14\ q + 12\ q^{2} + 4\ q^{3}$

$F_{6} = 2 + 18\ q + 24\ q^{2} + 16\ q^{3} + 4\ q^{4}$

$F_{7} = 2 + 22\ q + 40\ q^{2} + 40\ q^{3} + 20\ q^{4} + 4\ q^{5}$

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

bounce path

Description

The bounce path determined by an integer composition.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .