- FindStat

Identifier

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001313: Binary words ⟶ ℤ

Values

[1] => 10 => 01 => 2

[2] => 100 => 011 => 3

[1,1] => 110 => 001 => 3

[3] => 1000 => 0111 => 4

[2,1] => 1010 => 0101 => 5

[1,1,1] => 1110 => 0001 => 4

[4] => 10000 => 01111 => 5

[3,1] => 10010 => 01101 => 7

[2,2] => 1100 => 0011 => 6

[2,1,1] => 10110 => 01001 => 7

[1,1,1,1] => 11110 => 00001 => 5

[5] => 100000 => 011111 => 6

[4,1] => 100010 => 011101 => 9

[3,2] => 10100 => 01011 => 9

[3,1,1] => 100110 => 011001 => 10

[2,2,1] => 11010 => 00101 => 9

[2,1,1,1] => 101110 => 010001 => 9

[1,1,1,1,1] => 111110 => 000001 => 6

[6] => 1000000 => 0111111 => 7

[5,1] => 1000010 => 0111101 => 11

[4,2] => 100100 => 011011 => 12

[4,1,1] => 1000110 => 0111001 => 13

[3,3] => 11000 => 00111 => 10

[3,2,1] => 101010 => 010101 => 14

[3,1,1,1] => 1001110 => 0110001 => 13

[2,2,2] => 11100 => 00011 => 10

[2,2,1,1] => 110110 => 001001 => 12

[2,1,1,1,1] => 1011110 => 0100001 => 11

[1,1,1,1,1,1] => 1111110 => 0000001 => 7

[7] => 10000000 => 01111111 => 8

[6,1] => 10000010 => 01111101 => 13

[5,2] => 1000100 => 0111011 => 15

[5,1,1] => 10000110 => 01111001 => 16

[4,3] => 101000 => 010111 => 14

[4,2,1] => 1001010 => 0110101 => 19

[4,1,1,1] => 10001110 => 01110001 => 17

[3,3,1] => 110010 => 001101 => 16

[3,2,2] => 101100 => 010011 => 16

[3,2,1,1] => 1010110 => 0101001 => 19

[3,1,1,1,1] => 10011110 => 01100001 => 16

[2,2,2,1] => 111010 => 000101 => 14

[2,2,1,1,1] => 1101110 => 0010001 => 15

[2,1,1,1,1,1] => 10111110 => 01000001 => 13

[1,1,1,1,1,1,1] => 11111110 => 00000001 => 8

[8] => 100000000 => 011111111 => 9

[7,1] => 100000010 => 011111101 => 15

[6,2] => 10000100 => 01111011 => 18

[6,1,1] => 100000110 => 011111001 => 19

[5,3] => 1001000 => 0110111 => 18

[5,2,1] => 10001010 => 01110101 => 24

[5,1,1,1] => 100001110 => 011110001 => 21

[4,4] => 110000 => 001111 => 15

[4,3,1] => 1010010 => 0101101 => 23

[4,2,2] => 1001100 => 0110011 => 22

[4,2,1,1] => 10010110 => 01101001 => 26

[4,1,1,1,1] => 100011110 => 011100001 => 21

[3,3,2] => 110100 => 001011 => 19

[3,3,1,1] => 1100110 => 0011001 => 22

[3,2,2,1] => 1011010 => 0100101 => 23

[3,2,1,1,1] => 10101110 => 01010001 => 24

[3,1,1,1,1,1] => 100111110 => 011000001 => 19

[2,2,2,2] => 111100 => 000011 => 15

[2,2,2,1,1] => 1110110 => 0001001 => 18

[2,2,1,1,1,1] => 11011110 => 00100001 => 18

[2,1,1,1,1,1,1] => 101111110 => 010000001 => 15

[1,1,1,1,1,1,1,1] => 111111110 => 000000001 => 9

[7,2] => 100000100 => 011111011 => 21

[6,3] => 10001000 => 01110111 => 22

[6,2,1] => 100001010 => 011110101 => 29

[5,4] => 1010000 => 0101111 => 20

[5,3,1] => 10010010 => 01101101 => 30

[5,2,2] => 10001100 => 01110011 => 28

[5,2,1,1] => 100010110 => 011101001 => 33

[4,4,1] => 1100010 => 0011101 => 25

[4,3,2] => 1010100 => 0101011 => 28

[4,3,1,1] => 10100110 => 01011001 => 32

[4,2,2,1] => 10011010 => 01100101 => 32

[4,2,1,1,1] => 100101110 => 011010001 => 33

[3,3,3] => 111000 => 000111 => 20

[3,3,2,1] => 1101010 => 0010101 => 28

[3,3,1,1,1] => 11001110 => 00110001 => 28

[3,2,2,2] => 1011100 => 0100011 => 25

[3,2,2,1,1] => 10110110 => 01001001 => 30

[3,2,1,1,1,1] => 101011110 => 010100001 => 29

[2,2,2,2,1] => 1111010 => 0000101 => 20

[2,2,2,1,1,1] => 11101110 => 00010001 => 22

[2,2,1,1,1,1,1] => 110111110 => 001000001 => 21

[7,3] => 100001000 => 011110111 => 26

[6,4] => 10010000 => 01101111 => 25

[6,3,1] => 100010010 => 011101101 => 37

[6,2,2] => 100001100 => 011110011 => 34

[5,5] => 1100000 => 0011111 => 21

[5,4,1] => 10100010 => 01011101 => 34

[5,3,2] => 10010100 => 01101011 => 37

[5,3,1,1] => 100100110 => 011011001 => 42

[5,2,2,1] => 100011010 => 011100101 => 41

[4,4,2] => 1100100 => 0011011 => 31

[4,4,1,1] => 11000110 => 00111001 => 35

[4,3,3] => 1011000 => 0100111 => 30

[4,3,2,1] => 10101010 => 01010101 => 42

[4,3,1,1,1] => 101001110 => 010110001 => 41

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

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

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

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

$F_{5} = 2\ q^{6} + 4\ q^{9} + q^{10}$

$F_{6} = 2\ q^{7} + 2\ q^{10} + 2\ q^{11} + 2\ q^{12} + 2\ q^{13} + q^{14}$

$F_{7} = 2\ q^{8} + 2\ q^{13} + 2\ q^{14} + 2\ q^{15} + 4\ q^{16} + q^{17} + 2\ q^{19}$

$F_{8} = 2\ q^{9} + 4\ q^{15} + 4\ q^{18} + 3\ q^{19} + 2\ q^{21} + 2\ q^{22} + 2\ q^{23} + 2\ q^{24} + q^{26}$

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

complement

Description

Send a binary word to the word obtained by interchanging the two letters.

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.