- FindStat

Identifier

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

[1,3,2] => 01 => [2,1] => [1,1,0,0,1,0] => 2

[2,1,3] => 10 => [1,2] => [1,0,1,1,0,0] => 1

[2,3,1] => 10 => [1,2] => [1,0,1,1,0,0] => 1

[3,1,2] => 10 => [1,2] => [1,0,1,1,0,0] => 1

[3,2,1] => 11 => [1,1,1] => [1,0,1,0,1,0] => 3

[1,2,3,4] => 000 => [4] => [1,1,1,1,0,0,0,0] => 0

[1,2,4,3] => 001 => [3,1] => [1,1,1,0,0,0,1,0] => 3

[1,3,2,4] => 010 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,3,4,2] => 010 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,4,2,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0] => 2

[1,4,3,2] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 5

[2,1,3,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[2,1,4,3] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 4

[2,3,1,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[2,3,4,1] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[2,4,1,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[2,4,3,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 4

[3,1,2,4] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[3,1,4,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[3,2,1,4] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[3,2,4,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[3,4,1,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[3,4,2,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[4,1,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

[4,1,3,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[4,2,1,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[4,2,3,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3

[4,3,1,2] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 4

[4,3,2,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 6

[1,2,3,4,5] => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0

[1,2,3,5,4] => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4

[1,2,4,3,5] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[1,2,4,5,3] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[1,2,5,3,4] => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

[1,2,5,4,3] => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7

[1,3,2,4,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,3,2,5,4] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6

[1,3,4,2,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,3,4,5,2] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,3,5,2,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,3,5,4,2] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6

[1,4,2,3,5] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,4,2,5,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,4,3,2,5] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,4,3,5,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,4,5,2,3] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,4,5,3,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,5,2,3,4] => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,5,2,4,3] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,5,3,2,4] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,5,3,4,2] => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5

[1,5,4,2,3] => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6

[1,5,4,3,2] => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[2,1,3,4,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,1,3,5,4] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5

[2,1,4,3,5] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,1,4,5,3] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,1,5,3,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,1,5,4,3] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8

[2,3,1,4,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,3,1,5,4] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5

[2,3,4,1,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,3,4,5,1] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,3,5,1,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,3,5,4,1] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5

[2,4,1,3,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,4,1,5,3] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,4,3,1,5] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,4,3,5,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,4,5,1,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,4,5,3,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,5,1,3,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[2,5,1,4,3] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,5,3,1,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,5,3,4,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4

[2,5,4,1,3] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5

[2,5,4,3,1] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8

[3,1,2,4,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[3,1,2,5,4] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5

[3,1,4,2,5] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,1,4,5,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,1,5,2,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,1,5,4,2] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7

[3,2,1,4,5] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,2,1,5,4] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7

[3,2,4,1,5] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,2,4,5,1] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,2,5,1,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,2,5,4,1] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7

[3,4,1,2,5] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[3,4,1,5,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,4,2,1,5] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,4,2,5,1] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,4,5,1,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[3,4,5,2,1] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

[3,5,1,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

[3,5,1,4,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2} + q^{3}$

$F_{4} = 1 + 7\ q + 3\ q^{2} + 8\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + 15\ q + 7\ q^{2} + 34\ q^{3} + 18\ q^{4} + 14\ q^{5} + 18\ q^{6} + 8\ q^{7} + 3\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + 31\ q + 15\ q^{2} + 122\ q^{3} + 72\ q^{4} + 69\ q^{5} + 147\ q^{6} + 83\ q^{7} + 71\ q^{8} + 33\ q^{9} + 45\ q^{10} + 18\ q^{11} + 8\ q^{12} + 3\ q^{13} + q^{14} + q^{15}$

Description

The bounce statistic of a Dyck path.
The bounce path

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ .
The points where

$D'$ touches the

$x$ -axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ .
In particular, the bounce statistics of

$D$ and

$D'$ coincide.

Map

bounce path

Description

The bounce path determined by an integer composition.

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.

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.