- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 3

001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 3

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

011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0] => 3

100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

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

110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0] => 1

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

0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 4

0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 4

0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 4

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

0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 4

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

0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4

0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4

1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3

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

1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 3

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

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

1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

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

00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5

00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5

00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5

00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 5

00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5

00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 5

00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 5

00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5

01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5

01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 5

01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5

01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 5

01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 5

01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 5

01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5

01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5

10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4

10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 4

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

10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4

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

10101 => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4

10110 => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4

10111 => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

11000 => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3

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

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

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

11100 => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2

11101 => [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2

11110 => [1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

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

000000 => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 6

000001 => [6,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 6

000010 => [5,2] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 6

000011 => [5,1,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 6

000100 => [4,3] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 6

000101 => [4,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 6

000110 => [4,1,2] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 6

000111 => [4,1,1,1] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 6

001000 => [3,4] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 6

001001 => [3,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 6

001010 => [3,2,2] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 6

001011 => [3,2,1,1] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 6

001100 => [3,1,3] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 6

001101 => [3,1,2,1] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 6

001110 => [3,1,1,2] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 6

001111 => [3,1,1,1,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 6

010000 => [2,5] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 6

010001 => [2,4,1] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 6

010010 => [2,3,2] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 6

010011 => [2,3,1,1] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 6

010100 => [2,2,3] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 6

010101 => [2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 6

010110 => [2,2,1,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6

010111 => [2,2,1,1,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 6

011000 => [2,1,4] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 6

011001 => [2,1,3,1] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 6

011010 => [2,1,2,2] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6

011011 => [2,1,2,1,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 6

011100 => [2,1,1,3] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 6

011101 => [2,1,1,2,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 6

011110 => [2,1,1,1,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6

011111 => [2,1,1,1,1,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6

100000 => [1,6] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 5

100001 => [1,5,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 5

100010 => [1,4,2] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 5

100011 => [1,4,1,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 5

100100 => [1,3,3] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 5

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

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

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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

bounce path

Description

The bounce path determined by an integer composition.

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.

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.