- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤ

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] => 2

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] => 2

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] => 2

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] => 3

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] => 2

01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 3

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] => 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] => 2

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

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

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] => 3

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] => 2

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] => 2

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] => 3

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] => 2

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] => 2

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] => 3

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

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] => 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] => 4

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] => 2

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

011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 4

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] => 2

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

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] => 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] => 2

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

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

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] => 4

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

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Description

The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also St000684The global dimension of the LNakayama algebra associated to a Dyck path.. We associate the finitistic dominant dimension of the algebra to the corresponding 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 $w$.