- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000946: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

010 => [2,2] => [1,1,0,0,1,1,0,0] => 6

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

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

101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 5

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

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

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

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

0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 8

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

0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 6

0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 10

0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 7

0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3

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

1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 6

1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 9

1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 5

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

1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7

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

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

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

00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6

00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 10

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

00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 8

00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 13

00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 9

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

01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 6

01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 11

01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 15

01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 10

01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 7

01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 12

01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 8

01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 3

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

10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 7

10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 11

10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 6

10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 9

10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 14

10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5

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

11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8

11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 12

11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 7

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

11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 9

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

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

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

000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 7

000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 12

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

000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 10

000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 16

000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 11

000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 5

001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 8

001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 14

001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 19

001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 13

001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 9

001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 15

001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 10

001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 4

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

010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 12

010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 17

010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 11

010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 15

010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 21

010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 16

010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 10

011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 7

011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 13

011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 18

011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 12

011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 8

011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 14

011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 9

011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 3

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

100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 8

100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 13

100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 7

100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 11

100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 17

100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 12

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$F_{1} = 1 + q$

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

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

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

$F_{5} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + 3\ q^{8} + 3\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$

$F_{6} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 4\ q^{6} + 4\ q^{7} + 4\ q^{8} + 5\ q^{9} + 5\ q^{10} + 5\ q^{11} + 5\ q^{12} + 4\ q^{13} + 4\ q^{14} + 4\ q^{15} + 3\ q^{16} + 2\ q^{17} + 2\ q^{18} + q^{19} + q^{20} + q^{21}$

Description

The sum of the skew hook positions in a Dyck path.
A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step.
Write

$U_i$ for the

$i$ -th up step and

$D_j$ for the

$j$ -th down step in the Dyck path. Then the skew hook set is the set

$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$
This statistic is the sum of all elements in

$H$ .

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.