- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤ

Values

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

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

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

01 => [2,1] => [1,1] => [1,0,1,0] => 3

10 => [1,2] => [1,1] => [1,0,1,0] => 3

11 => [1,1,1] => [3] => [1,1,1,0,0,0] => 4

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

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

010 => [2,2] => [2] => [1,1,0,0] => 3

011 => [2,1,1] => [1,2] => [1,0,1,1,0,0] => 4

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

101 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0] => 4

110 => [1,1,2] => [2,1] => [1,1,0,0,1,0] => 4

111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => 5

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

0001 => [4,1] => [1,1] => [1,0,1,0] => 3

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

0011 => [3,1,1] => [1,2] => [1,0,1,1,0,0] => 4

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

0101 => [2,2,1] => [2,1] => [1,1,0,0,1,0] => 4

0110 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0] => 4

0111 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 5

1000 => [1,4] => [1,1] => [1,0,1,0] => 3

1001 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0] => 4

1010 => [1,2,2] => [1,2] => [1,0,1,1,0,0] => 4

1011 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

1100 => [1,1,3] => [2,1] => [1,1,0,0,1,0] => 4

1101 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 5

1110 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0] => 5

1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 6

00000 => [6] => [1] => [1,0] => 2

00001 => [5,1] => [1,1] => [1,0,1,0] => 3

00010 => [4,2] => [1,1] => [1,0,1,0] => 3

00011 => [4,1,1] => [1,2] => [1,0,1,1,0,0] => 4

00100 => [3,3] => [2] => [1,1,0,0] => 3

00101 => [3,2,1] => [1,1,1] => [1,0,1,0,1,0] => 4

00110 => [3,1,2] => [1,1,1] => [1,0,1,0,1,0] => 4

00111 => [3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 5

01000 => [2,4] => [1,1] => [1,0,1,0] => 3

01001 => [2,3,1] => [1,1,1] => [1,0,1,0,1,0] => 4

01010 => [2,2,2] => [3] => [1,1,1,0,0,0] => 4

01011 => [2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0] => 5

01100 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0] => 4

01101 => [2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

01110 => [2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 5

01111 => [2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 6

10000 => [1,5] => [1,1] => [1,0,1,0] => 3

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

10010 => [1,3,2] => [1,1,1] => [1,0,1,0,1,0] => 4

10011 => [1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

10100 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0] => 4

10101 => [1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 5

10110 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

10111 => [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 6

11000 => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 4

11001 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 5

11010 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 5

11011 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 6

11100 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 5

11101 => [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 6

11110 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 6

11111 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7

000000 => [7] => [1] => [1,0] => 2

000001 => [6,1] => [1,1] => [1,0,1,0] => 3

000010 => [5,2] => [1,1] => [1,0,1,0] => 3

000011 => [5,1,1] => [1,2] => [1,0,1,1,0,0] => 4

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

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

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

000111 => [4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 5

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

001001 => [3,3,1] => [2,1] => [1,1,0,0,1,0] => 4

001010 => [3,2,2] => [1,2] => [1,0,1,1,0,0] => 4

001011 => [3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

001100 => [3,1,3] => [1,1,1] => [1,0,1,0,1,0] => 4

001101 => [3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

001110 => [3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 5

001111 => [3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 6

010000 => [2,5] => [1,1] => [1,0,1,0] => 3

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

010010 => [2,3,2] => [1,1,1] => [1,0,1,0,1,0] => 4

010011 => [2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

010100 => [2,2,3] => [2,1] => [1,1,0,0,1,0] => 4

010101 => [2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0] => 5

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

010111 => [2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 6

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

011001 => [2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

011010 => [2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

011011 => [2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 6

011100 => [2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0] => 5

011101 => [2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 6

011110 => [2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 6

011111 => [2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 7

100000 => [1,6] => [1,1] => [1,0,1,0] => 3

100001 => [1,5,1] => [1,1,1] => [1,0,1,0,1,0] => 4

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

100011 => [1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 5

100100 => [1,3,3] => [1,2] => [1,0,1,1,0,0] => 4

100101 => [1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

100110 => [1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 5

>>> Load all 1671 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

$F_{1} = q^{2} + q^{3}$

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

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

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

$F_{5} = q^{2} + 5\ q^{3} + 10\ q^{4} + 10\ q^{5} + 5\ q^{6} + q^{7}$

$F_{6} = q^{2} + 6\ q^{3} + 15\ q^{4} + 20\ q^{5} + 15\ q^{6} + 6\ q^{7} + q^{8}$

Description

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

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

$C$ .

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.