- FindStat

Identifier

Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤ

Values

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

1 => 1 => [1] => 1

00 => 00 => [2] => 1

01 => 10 => [1,1] => 1

10 => 01 => [1,1] => 1

11 => 11 => [2] => 1

000 => 000 => [3] => 1

001 => 100 => [1,2] => 1

010 => 001 => [2,1] => 2

011 => 101 => [1,1,1] => 1

100 => 010 => [1,1,1] => 1

101 => 110 => [2,1] => 2

110 => 011 => [1,2] => 1

111 => 111 => [3] => 1

0000 => 0000 => [4] => 1

0001 => 1000 => [1,3] => 1

0010 => 0001 => [3,1] => 3

0011 => 1001 => [1,2,1] => 2

0100 => 0010 => [2,1,1] => 3

0101 => 1010 => [1,1,1,1] => 1

0110 => 0011 => [2,2] => 3

0111 => 1011 => [1,1,2] => 1

1000 => 0100 => [1,1,2] => 1

1001 => 1100 => [2,2] => 3

1010 => 0101 => [1,1,1,1] => 1

1011 => 1101 => [2,1,1] => 3

1100 => 0110 => [1,2,1] => 2

1101 => 1110 => [3,1] => 3

1110 => 0111 => [1,3] => 1

1111 => 1111 => [4] => 1

00000 => 00000 => [5] => 1

00001 => 10000 => [1,4] => 1

00010 => 00001 => [4,1] => 4

00011 => 10001 => [1,3,1] => 3

00100 => 00010 => [3,1,1] => 6

00101 => 10010 => [1,2,1,1] => 3

00110 => 00011 => [3,2] => 6

00111 => 10011 => [1,2,2] => 3

01000 => 00100 => [2,1,2] => 4

01001 => 10100 => [1,1,1,2] => 1

01010 => 00101 => [2,1,1,1] => 4

01011 => 10101 => [1,1,1,1,1] => 1

01100 => 00110 => [2,2,1] => 8

01101 => 10110 => [1,1,2,1] => 2

01110 => 00111 => [2,3] => 4

01111 => 10111 => [1,1,3] => 1

10000 => 01000 => [1,1,3] => 1

10001 => 11000 => [2,3] => 4

10010 => 01001 => [1,1,2,1] => 2

10011 => 11001 => [2,2,1] => 8

10100 => 01010 => [1,1,1,1,1] => 1

10101 => 11010 => [2,1,1,1] => 4

10110 => 01011 => [1,1,1,2] => 1

10111 => 11011 => [2,1,2] => 4

11000 => 01100 => [1,2,2] => 3

11001 => 11100 => [3,2] => 6

11010 => 01101 => [1,2,1,1] => 3

11011 => 11101 => [3,1,1] => 6

11100 => 01110 => [1,3,1] => 3

11101 => 11110 => [4,1] => 4

11110 => 01111 => [1,4] => 1

11111 => 11111 => [5] => 1

000000 => 000000 => [6] => 1

000001 => 100000 => [1,5] => 1

000010 => 000001 => [5,1] => 5

000011 => 100001 => [1,4,1] => 4

000100 => 000010 => [4,1,1] => 10

000101 => 100010 => [1,3,1,1] => 6

000110 => 000011 => [4,2] => 10

000111 => 100011 => [1,3,2] => 6

001000 => 000100 => [3,1,2] => 10

001001 => 100100 => [1,2,1,2] => 4

001010 => 000101 => [3,1,1,1] => 10

001011 => 100101 => [1,2,1,1,1] => 4

001100 => 000110 => [3,2,1] => 20

001101 => 100110 => [1,2,2,1] => 8

001110 => 000111 => [3,3] => 10

001111 => 100111 => [1,2,3] => 4

010000 => 001000 => [2,1,3] => 5

010001 => 101000 => [1,1,1,3] => 1

010010 => 001001 => [2,1,2,1] => 10

010011 => 101001 => [1,1,1,2,1] => 2

010100 => 001010 => [2,1,1,1,1] => 5

010101 => 101010 => [1,1,1,1,1,1] => 1

010110 => 001011 => [2,1,1,2] => 5

010111 => 101011 => [1,1,1,1,2] => 1

011000 => 001100 => [2,2,2] => 15

011001 => 101100 => [1,1,2,2] => 3

011010 => 001101 => [2,2,1,1] => 15

011011 => 101101 => [1,1,2,1,1] => 3

011100 => 001110 => [2,3,1] => 15

011101 => 101110 => [1,1,3,1] => 3

011110 => 001111 => [2,4] => 5

011111 => 101111 => [1,1,4] => 1

100000 => 010000 => [1,1,4] => 1

100001 => 110000 => [2,4] => 5

100010 => 010001 => [1,1,3,1] => 3

100011 => 110001 => [2,3,1] => 15

100100 => 010010 => [1,1,2,1,1] => 3

100101 => 110010 => [2,2,1,1] => 15

100110 => 010011 => [1,1,2,2] => 3

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Description

The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.

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 $w$.

Map

rotate back-to-front

Description

The rotation of a binary word, last letter first.
This is the word obtained by moving the last letter to the beginnig.