- FindStat

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Identifier

St000420: Dyck paths ⟶ ℤ (values match St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => 42

[1,0,1,0,1,0,1,1,0,0] => 28

[1,0,1,0,1,1,0,0,1,0] => 32

[1,0,1,0,1,1,0,1,0,0] => 23

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

[1,0,1,1,0,0,1,0,1,0] => 32

[1,0,1,1,0,0,1,1,0,0] => 22

[1,0,1,1,0,1,0,0,1,0] => 26

[1,0,1,1,0,1,0,1,0,0] => 19

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

[1,0,1,1,1,0,0,0,1,0] => 17

[1,0,1,1,1,0,0,1,0,0] => 13

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

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

[1,1,0,0,1,0,1,0,1,0] => 28

[1,1,0,0,1,0,1,1,0,0] => 19

[1,1,0,0,1,1,0,0,1,0] => 22

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

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

[1,1,0,1,0,0,1,0,1,0] => 23

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

[1,1,0,1,0,1,0,0,1,0] => 19

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

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

[1,1,0,1,1,0,0,0,1,0] => 13

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => 132

[1,0,1,0,1,0,1,0,1,1,0,0] => 90

[1,0,1,0,1,0,1,1,0,0,1,0] => 104

[1,0,1,0,1,0,1,1,0,1,0,0] => 76

[1,0,1,0,1,0,1,1,1,0,0,0] => 48

[1,0,1,0,1,1,0,0,1,0,1,0] => 107

[1,0,1,0,1,1,0,0,1,1,0,0] => 75

[1,0,1,0,1,1,0,1,0,0,1,0] => 89

[1,0,1,0,1,1,0,1,0,1,0,0] => 66

[1,0,1,0,1,1,0,1,1,0,0,0] => 43

[1,0,1,0,1,1,1,0,0,0,1,0] => 62

[1,0,1,0,1,1,1,0,0,1,0,0] => 48

[1,0,1,0,1,1,1,0,1,0,0,0] => 34

[1,0,1,0,1,1,1,1,0,0,0,0] => 20

[1,0,1,1,0,0,1,0,1,0,1,0] => 104

[1,0,1,1,0,0,1,0,1,1,0,0] => 72

[1,0,1,1,0,0,1,1,0,0,1,0] => 84

[1,0,1,1,0,0,1,1,0,1,0,0] => 62

[1,0,1,1,0,0,1,1,1,0,0,0] => 40

[1,0,1,1,0,1,0,0,1,0,1,0] => 89

[1,0,1,1,0,1,0,0,1,1,0,0] => 63

[1,0,1,1,0,1,0,1,0,0,1,0] => 75

[1,0,1,1,0,1,0,1,0,1,0,0] => 56

[1,0,1,1,0,1,0,1,1,0,0,0] => 37

[1,0,1,1,0,1,1,0,0,0,1,0] => 54

[1,0,1,1,0,1,1,0,0,1,0,0] => 42

[1,0,1,1,0,1,1,0,1,0,0,0] => 30

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

[1,0,1,1,1,0,0,0,1,0,1,0] => 62

[1,0,1,1,1,0,0,0,1,1,0,0] => 45

[1,0,1,1,1,0,0,1,0,0,1,0] => 54

[1,0,1,1,1,0,0,1,0,1,0,0] => 41

[1,0,1,1,1,0,0,1,1,0,0,0] => 28

[1,0,1,1,1,0,1,0,0,0,1,0] => 42

[1,0,1,1,1,0,1,0,0,1,0,0] => 33

[1,0,1,1,1,0,1,0,1,0,0,0] => 24

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

[1,0,1,1,1,1,0,0,0,0,1,0] => 26

>>> Load all 1200 entries. <<<

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

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

$F_{4} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + q^{6} + 2\ q^{7} + 2\ q^{9} + q^{10} + q^{14}$

$F_{5} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + q^{6} + 2\ q^{7} + 4\ q^{9} + 3\ q^{10} + 2\ q^{12} + 2\ q^{13} + 3\ q^{14} + 2\ q^{16} + q^{17} + 3\ q^{19} + 2\ q^{22} + 2\ q^{23} + q^{26} + 2\ q^{28} + 2\ q^{32} + q^{42}$

$F_{6} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 2\ q^{7} + 4\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + 2\ q^{13} + 3\ q^{14} + 4\ q^{15} + 4\ q^{16} + q^{17} + 2\ q^{18} + 3\ q^{19} + 3\ q^{20} + 2\ q^{21} + 2\ q^{22} + 2\ q^{23} + 2\ q^{24} + 2\ q^{25} + 2\ q^{26} + 4\ q^{28} + 4\ q^{30} + 2\ q^{31} + 2\ q^{32} + 2\ q^{33} + 4\ q^{34} + 2\ q^{35} + 2\ q^{37} + 2\ q^{40} + 2\ q^{41} + 4\ q^{42} + 2\ q^{43} + 2\ q^{45} + q^{46} + 2\ q^{47} + 4\ q^{48} + 3\ q^{53} + 2\ q^{54} + 2\ q^{56} + 5\ q^{62} + 2\ q^{63} + 2\ q^{66} + 2\ q^{72} + 3\ q^{75} + 2\ q^{76} + q^{84} + 2\ q^{89} + 2\ q^{90} + 2\ q^{104} + q^{107} + q^{132}$

$F_{7} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + 4\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + 4\ q^{13} + 3\ q^{14} + 4\ q^{15} + 4\ q^{16} + q^{17} + 4\ q^{18} + 5\ q^{19} + 3\ q^{20} + 4\ q^{21} + 4\ q^{22} + 2\ q^{23} + 2\ q^{24} + 6\ q^{25} + 2\ q^{26} + 2\ q^{27} + 4\ q^{28} + 2\ q^{29} + 4\ q^{30} + 4\ q^{31} + 2\ q^{32} + 2\ q^{33} + 6\ q^{34} + 4\ q^{35} + 2\ q^{36} + 5\ q^{37} + 4\ q^{40} + 2\ q^{41} + 4\ q^{42} + 4\ q^{43} + 2\ q^{45} + 5\ q^{46} + 4\ q^{47} + 4\ q^{48} + 6\ q^{50} + 4\ q^{51} + 4\ q^{52} + 3\ q^{53} + 2\ q^{54} + 8\ q^{55} + 2\ q^{56} + 4\ q^{61} + 8\ q^{62} + 2\ q^{63} + 4\ q^{65} + 6\ q^{66} + 4\ q^{67} + q^{69} + 4\ q^{70} + 2\ q^{71} + 2\ q^{72} + 7\ q^{75} + 4\ q^{76} + 2\ q^{78} + 4\ q^{79} + 2\ q^{80} + 4\ q^{81} + 2\ q^{82} + 2\ q^{83} + q^{84} + 2\ q^{85} + 2\ q^{87} + 4\ q^{89} + 2\ q^{90} + 2\ q^{91} + 2\ q^{94} + 6\ q^{95} + 4\ q^{96} + 4\ q^{97} + 4\ q^{99} + 2\ q^{103} + 2\ q^{104} + 4\ q^{105} + 2\ q^{106} + 3\ q^{107} + 2\ q^{108} + 2\ q^{109} + 2\ q^{110} + 2\ q^{112} + 2\ q^{113} + 2\ q^{114} + 2\ q^{115} + 2\ q^{116} + 2\ q^{117} + 3\ q^{118} + 2\ q^{121} + 4\ q^{123} + 2\ q^{125} + q^{126} + 2\ q^{127} + 2\ q^{129} + 3\ q^{132} + q^{134} + 4\ q^{136} + 2\ q^{137} + 2\ q^{139} + 2\ q^{140} + 2\ q^{142} + 8\ q^{145} + 2\ q^{146} + 4\ q^{151} + 3\ q^{156} + 4\ q^{157} + 2\ q^{158} + 2\ q^{161} + q^{162} + 2\ q^{163} + 2\ q^{165} + 4\ q^{171} + 2\ q^{173} + 2\ q^{174} + 2\ q^{179} + 2\ q^{180} + q^{181} + 2\ q^{185} + 2\ q^{188} + 2\ q^{189} + 2\ q^{193} + 2\ q^{199} + 2\ q^{202} + 2\ q^{205} + q^{207} + 2\ q^{209} + 2\ q^{211} + 2\ q^{214} + 2\ q^{219} + 2\ q^{220} + 2\ q^{225} + 2\ q^{227} + q^{230} + q^{233} + 2\ q^{241} + 2\ q^{251} + 2\ q^{252} + 4\ q^{255} + 2\ q^{268} + q^{281} + 2\ q^{295} + 2\ q^{297} + 2\ q^{303} + q^{314} + 2\ q^{345} + 2\ q^{359} + q^{429}$

Description

The number of Dyck paths that are weakly above a Dyck path.

Code

def contains(A,B):
     Aa = A.to_area_sequence()
     Bb = B.to_area_sequence()
     return all( Aa[i] >= Bb[i] for i in range(len(Aa)) )

def statistic(D):
    n = len(D)/2
    return sum( 1 for A in DyckWords(n) if contains(A,D) )

Created

Feb 26, 2016 at 10:36 by Christian Stump

Updated

Feb 26, 2016 at 10:36 by Christian Stump