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Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000047: Integer compositions ⟶ ℤ

Values

0 => [1] => 1

1 => [1] => 1

00 => [2] => 1

01 => [1,1] => 1

10 => [1,1] => 1

11 => [2] => 1

000 => [3] => 1

001 => [2,1] => 2

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

011 => [1,2] => 1

100 => [1,2] => 1

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

110 => [2,1] => 2

111 => [3] => 1

0000 => [4] => 1

0001 => [3,1] => 3

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

0011 => [2,2] => 3

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

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

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

0111 => [1,3] => 1

1000 => [1,3] => 1

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

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

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

1100 => [2,2] => 3

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

1110 => [3,1] => 3

1111 => [4] => 1

00000 => [5] => 1

00001 => [4,1] => 4

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

00011 => [3,2] => 6

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

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

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

00111 => [2,3] => 4

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

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

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

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

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

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

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

01111 => [1,4] => 1

10000 => [1,4] => 1

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

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

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

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

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

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

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

11000 => [2,3] => 4

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

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

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

11100 => [3,2] => 6

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

11110 => [4,1] => 4

11111 => [5] => 1

000000 => [6] => 1

000001 => [5,1] => 5

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

000011 => [4,2] => 10

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

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

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

000111 => [3,3] => 10

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

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

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

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

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

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

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

001111 => [2,4] => 5

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

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

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

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

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

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

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

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

011000 => [1,2,3] => 4

011001 => [1,2,2,1] => 8

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

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

011100 => [1,3,2] => 6

011101 => [1,3,1,1] => 6

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

011111 => [1,5] => 1

100000 => [1,5] => 1

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

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

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

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

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

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

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$F_{1} = 2\ q$

$F_{2} = 4\ q$

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

$F_{4} = 8\ q + 2\ q^{2} + 6\ q^{3}$

$F_{5} = 10\ q + 2\ q^{2} + 6\ q^{3} + 8\ q^{4} + 4\ q^{6} + 2\ q^{8}$

$F_{6} = 12\ q + 2\ q^{2} + 6\ q^{3} + 8\ q^{4} + 10\ q^{5} + 4\ q^{6} + 2\ q^{8} + 12\ q^{10} + 6\ q^{15} + 2\ q^{20}$

$F_{7} = 14\ q + 2\ q^{2} + 6\ q^{3} + 8\ q^{4} + 10\ q^{5} + 16\ q^{6} + 2\ q^{8} + 12\ q^{10} + 2\ q^{12} + 18\ q^{15} + 6\ q^{18} + 8\ q^{20} + 8\ q^{24} + 2\ q^{30} + 4\ q^{36} + 2\ q^{40} + 6\ q^{45} + 2\ q^{48}$

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

$w$ .