- FindStat

Identifier

Mp00224: Binary words —runsort⟶ Binary words
St001313: Binary words ⟶ ℤ

Values

0 => 0 => 1

1 => 1 => 1

00 => 00 => 1

01 => 01 => 2

10 => 01 => 2

11 => 11 => 1

000 => 000 => 1

001 => 001 => 3

010 => 001 => 3

011 => 011 => 3

100 => 001 => 3

101 => 011 => 3

110 => 011 => 3

111 => 111 => 1

0000 => 0000 => 1

0001 => 0001 => 4

0010 => 0001 => 4

0011 => 0011 => 6

0100 => 0001 => 4

0101 => 0101 => 5

0110 => 0011 => 6

0111 => 0111 => 4

1000 => 0001 => 4

1001 => 0011 => 6

1010 => 0011 => 6

1011 => 0111 => 4

1100 => 0011 => 6

1101 => 0111 => 4

1110 => 0111 => 4

1111 => 1111 => 1

00000 => 00000 => 1

00001 => 00001 => 5

00010 => 00001 => 5

00011 => 00011 => 10

00100 => 00001 => 5

00101 => 00101 => 9

00110 => 00011 => 10

00111 => 00111 => 10

01000 => 00001 => 5

01001 => 00101 => 9

01010 => 00101 => 9

01011 => 01011 => 9

01100 => 00011 => 10

01101 => 01011 => 9

01110 => 00111 => 10

01111 => 01111 => 5

10000 => 00001 => 5

10001 => 00011 => 10

10010 => 00011 => 10

10011 => 00111 => 10

10100 => 00011 => 10

10101 => 01011 => 9

10110 => 00111 => 10

10111 => 01111 => 5

11000 => 00011 => 10

11001 => 00111 => 10

11010 => 00111 => 10

11011 => 01111 => 5

11100 => 00111 => 10

11101 => 01111 => 5

11110 => 01111 => 5

11111 => 11111 => 1

000000 => 000000 => 1

000001 => 000001 => 6

000010 => 000001 => 6

000011 => 000011 => 15

000100 => 000001 => 6

000101 => 000101 => 14

000110 => 000011 => 15

000111 => 000111 => 20

001000 => 000001 => 6

001001 => 001001 => 12

001010 => 000101 => 14

001011 => 001011 => 19

001100 => 000011 => 15

001101 => 001101 => 16

001110 => 000111 => 20

001111 => 001111 => 15

010000 => 000001 => 6

010001 => 000101 => 14

010010 => 000101 => 14

010011 => 001101 => 16

010100 => 000101 => 14

010101 => 010101 => 14

010110 => 001011 => 19

010111 => 010111 => 14

011000 => 000011 => 15

011001 => 001011 => 19

011010 => 001011 => 19

011011 => 011011 => 12

011100 => 000111 => 20

011101 => 010111 => 14

011110 => 001111 => 15

011111 => 011111 => 6

100000 => 000001 => 6

100001 => 000011 => 15

100010 => 000011 => 15

100011 => 000111 => 20

100100 => 000011 => 15

100101 => 001011 => 19

100110 => 000111 => 20

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$F_{1} = 2\ q$

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

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

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

$F_{5} = 2\ q + 10\ q^{5} + 6\ q^{9} + 14\ q^{10}$

$F_{6} = 2\ q + 12\ q^{6} + 2\ q^{12} + 11\ q^{14} + 18\ q^{15} + 2\ q^{16} + 7\ q^{19} + 10\ q^{20}$

$F_{7} = 2\ q + 14\ q^{7} + 6\ q^{18} + 14\ q^{20} + 22\ q^{21} + 2\ q^{25} + 8\ q^{28} + 4\ q^{30} + 8\ q^{31} + 22\ q^{34} + 26\ q^{35}$

$F_{8} = 2\ q + 16\ q^{8} + 2\ q^{22} + 10\ q^{25} + 18\ q^{27} + 26\ q^{28} + 2\ q^{36} + 2\ q^{40} + q^{42} + 6\ q^{43} + 4\ q^{46} + 3\ q^{47} + 14\ q^{48} + 6\ q^{50} + 16\ q^{52} + q^{53} + 34\ q^{55} + 32\ q^{56} + 13\ q^{62} + 14\ q^{65} + 17\ q^{69} + 17\ q^{70}$

$F_{9} = 2\ q + 18\ q^{9} + 6\ q^{30} + 14\ q^{33} + 22\ q^{35} + 30\ q^{36} + 2\ q^{49} + 2\ q^{55} + 4\ q^{64} + 4\ q^{65} + 18\ q^{70} + 3\ q^{71} + 10\ q^{74} + 22\ q^{75} + 8\ q^{77} + 24\ q^{80} + 38\ q^{83} + 38\ q^{84} + 10\ q^{90} + 4\ q^{91} + 6\ q^{95} + 6\ q^{96} + 9\ q^{99} + 14\ q^{105} + 12\ q^{108} + 8\ q^{111} + 44\ q^{117} + 20\ q^{120} + 26\ q^{121} + 46\ q^{125} + 42\ q^{126}$

Description

The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating

$1$ 's as steps

$(1,0)$ and

$0$ 's as steps

$(0,1)$ . Given a binary word

$w$ , this statistic counts the number of lattice paths from the origin to the same endpoint as

$w$ that stay weakly above

$w$ .
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.

Map

runsort

Description

The word obtained by sorting the weakly increasing runs lexicographically.