- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

010 => [1,1,1] => [1,0,1,0,1,0] => 3

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

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

101 => [1,1,1] => [1,0,1,0,1,0] => 3

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

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] => 2

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

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

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

0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

0111 => [1,3] => [1,0,1,1,1,0,0,0] => 2

1000 => [1,3] => [1,0,1,1,1,0,0,0] => 2

1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => 3

1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4

1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

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

1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

1110 => [3,1] => [1,1,1,0,0,0,1,0] => 2

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] => 2

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

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] => 3

00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

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] => 3

01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5

01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 4

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] => 4

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

01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

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

10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

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] => 4

10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 5

10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

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

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] => 4

11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3

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] => 3

11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2

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] => 2

000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 3

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] => 3

000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 4

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] => 3

001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 4

001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5

001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4

001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3

001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 4

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] => 3

010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4

010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5

010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4

010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5

010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6

010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5

010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4

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] => 5

011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4

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] => 4

011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3

011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 3

100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4

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] => 4

100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5

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

>>> Load all 126 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

$F_{1} = 2\ q$

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

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

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

$F_{5} = 2\ q + 8\ q^{2} + 12\ q^{3} + 8\ q^{4} + 2\ q^{5}$

$F_{6} = 2\ q + 10\ q^{2} + 20\ q^{3} + 20\ q^{4} + 10\ q^{5} + 2\ q^{6}$

Description

The number of peaks of a Dyck path.

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

$w$ .