- FindStat

Identifier

Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤ

Values

[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

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

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$F_{2} = 1 + q$

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

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

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

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

$F_{7} = 1 + 63\ q + 31\ q^{2} + 406\ q^{3} + 252\ q^{4} + 263\ q^{5} + 822\ q^{6} + 544\ q^{7} + 554\ q^{8} + 399\ q^{9} + 613\ q^{10} + 351\ q^{11} + 307\ q^{12} + 160\ q^{13} + 102\ q^{14} + 96\ q^{15} + 45\ q^{16} + 18\ q^{17} + 8\ q^{18} + 3\ q^{19} + q^{20} + q^{21}$

Description

The major index east count of a Dyck path.
The descent set

$\operatorname{des}(D)$ of a Dyck path

$D = D_1 \cdots D_{2n}$ with

$D_i \in \{N,E\}$ is given by all indices

$i$ such that

$D_i = E$ and

$D_{i+1} = N$ . This is, the positions of the valleys of

$D$ .
The major index of a Dyck path is then the sum of the positions of the valleys,

$\sum_{i \in \operatorname{des}(D)} i$ , see St000027The major index of a Dyck path..
The major index east count is given by

$\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$ .

Map

descent bottoms

Description

The descent bottoms of a permutation as a binary word.

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

bounce path

Description

The bounce path determined by an integer composition.