- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

111 => [3] => [3] => [1,1,1,0,0,0] => 0

0000 => [4] => [4] => [1,1,1,1,0,0,0,0] => 0

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

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

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

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

0101 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

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

1010 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

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

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

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

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

1111 => [4] => [4] => [1,1,1,1,0,0,0,0] => 0

00000 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0

00001 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

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

00011 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

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

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

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

00111 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

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

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

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

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

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

01101 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

01110 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

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

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

10001 => [1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

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

10011 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 4

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

10101 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4

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

10111 => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4

11000 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

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

11010 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3

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

11100 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

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

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

11111 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0

000000 => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0

000001 => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

000010 => [4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2

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

000100 => [3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3

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

000110 => [3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

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

001000 => [2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4

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

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

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

001100 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

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

001110 => [2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4

001111 => [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

010000 => [1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5

010001 => [1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5

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

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

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

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

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

010111 => [1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5

011000 => [1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5

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

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

011011 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5

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

011101 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5

011110 => [1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5

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

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

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

100010 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5

100011 => [1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5

100100 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5

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

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

100111 => [1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5

101000 => [1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5

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

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

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

$F_{5} = 2 + 2\ q + 4\ q^{2} + 8\ q^{3} + 16\ q^{4}$

$F_{6} = 2 + 2\ q + 4\ q^{2} + 8\ q^{3} + 16\ q^{4} + 32\ q^{5}$

$F_{7} = 2 + 2\ q + 4\ q^{2} + 8\ q^{3} + 16\ q^{4} + 32\ q^{5} + 64\ q^{6}$

Description

The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley

$v$ in a Dyck path

$D$ there is a corresponding tunnel, which
is the factor

$T_v = s_i\dots s_j$ of

$D$ where

$s_i$ is the step after the first intersection of

$D$ with the line

$y = ht(v)$ to the left of

$s_j$ . This statistic is

$\sum_v (j_v-i_v)/2.$

Map

reverse

Description

Return the reversal of a composition.
That is, the composition

$(i_1, i_2, \ldots, i_k)$ is sent to

$(i_k, i_{k-1}, \ldots, i_1)$ .

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

bounce path

Description

The bounce path determined by an integer composition.