- FindStat

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Identifier

St000419: Dyck paths ⟶ ℤ (values match St000420The number of Dyck paths that are weakly above a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,1,0,0] => 27

[1,0,1,0,1,1,0,0,1,0] => 31

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

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

[1,0,1,1,0,0,1,0,1,0] => 31

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

[1,0,1,1,0,1,0,0,1,0] => 25

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

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => 131

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

[1,0,1,0,1,0,1,1,0,0,1,0] => 103

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

[1,0,1,0,1,0,1,1,1,0,0,0] => 47

[1,0,1,0,1,1,0,0,1,0,1,0] => 106

[1,0,1,0,1,1,0,0,1,1,0,0] => 74

[1,0,1,0,1,1,0,1,0,0,1,0] => 88

[1,0,1,0,1,1,0,1,0,1,0,0] => 65

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

[1,0,1,0,1,1,1,0,0,0,1,0] => 61

[1,0,1,0,1,1,1,0,0,1,0,0] => 47

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

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

[1,0,1,1,0,0,1,0,1,0,1,0] => 103

[1,0,1,1,0,0,1,0,1,1,0,0] => 71

[1,0,1,1,0,0,1,1,0,0,1,0] => 83

[1,0,1,1,0,0,1,1,0,1,0,0] => 61

[1,0,1,1,0,0,1,1,1,0,0,0] => 39

[1,0,1,1,0,1,0,0,1,0,1,0] => 88

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

[1,0,1,1,0,1,0,1,0,0,1,0] => 74

[1,0,1,1,0,1,0,1,0,1,0,0] => 55

[1,0,1,1,0,1,0,1,1,0,0,0] => 36

[1,0,1,1,0,1,1,0,0,0,1,0] => 53

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

[1,0,1,1,0,1,1,0,1,0,0,0] => 29

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

[1,0,1,1,1,0,0,0,1,0,1,0] => 61

[1,0,1,1,1,0,0,0,1,1,0,0] => 44

[1,0,1,1,1,0,0,1,0,0,1,0] => 53

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

[1,0,1,1,1,0,0,1,1,0,0,0] => 27

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

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

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

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

[1,0,1,1,1,1,0,0,0,0,1,0] => 25

>>> Load all 1200 entries. <<<

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$F_{2} = 1 + q$

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

$F_{4} = 1 + q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + 2\ q^{8} + q^{9} + q^{13}$

$F_{5} = 1 + q + 2\ q^{2} + 2\ q^{3} + 3\ q^{4} + q^{5} + 2\ q^{6} + 4\ q^{8} + 3\ q^{9} + 2\ q^{11} + 2\ q^{12} + 3\ q^{13} + 2\ q^{15} + q^{16} + 3\ q^{18} + 2\ q^{21} + 2\ q^{22} + q^{25} + 2\ q^{27} + 2\ q^{31} + q^{41}$

$F_{6} = 1 + q + 2\ q^{2} + 2\ q^{3} + 3\ q^{4} + 3\ q^{5} + 2\ q^{6} + 4\ q^{8} + 3\ q^{9} + 2\ q^{10} + 2\ q^{11} + 2\ q^{12} + 3\ q^{13} + 4\ q^{14} + 4\ q^{15} + q^{16} + 2\ q^{17} + 3\ q^{18} + 3\ q^{19} + 2\ q^{20} + 2\ q^{21} + 2\ q^{22} + 2\ q^{23} + 2\ q^{24} + 2\ q^{25} + 4\ q^{27} + 4\ q^{29} + 2\ q^{30} + 2\ q^{31} + 2\ q^{32} + 4\ q^{33} + 2\ q^{34} + 2\ q^{36} + 2\ q^{39} + 2\ q^{40} + 4\ q^{41} + 2\ q^{42} + 2\ q^{44} + q^{45} + 2\ q^{46} + 4\ q^{47} + 3\ q^{52} + 2\ q^{53} + 2\ q^{55} + 5\ q^{61} + 2\ q^{62} + 2\ q^{65} + 2\ q^{71} + 3\ q^{74} + 2\ q^{75} + q^{83} + 2\ q^{88} + 2\ q^{89} + 2\ q^{103} + q^{106} + q^{131}$

$F_{7} = 1 + q + 2\ q^{2} + 2\ q^{3} + 3\ q^{4} + 3\ q^{5} + 4\ q^{6} + 4\ q^{8} + 3\ q^{9} + 2\ q^{10} + 2\ q^{11} + 4\ q^{12} + 3\ q^{13} + 4\ q^{14} + 4\ q^{15} + q^{16} + 4\ q^{17} + 5\ q^{18} + 3\ q^{19} + 4\ q^{20} + 4\ q^{21} + 2\ q^{22} + 2\ q^{23} + 6\ q^{24} + 2\ q^{25} + 2\ q^{26} + 4\ q^{27} + 2\ q^{28} + 4\ q^{29} + 4\ q^{30} + 2\ q^{31} + 2\ q^{32} + 6\ q^{33} + 4\ q^{34} + 2\ q^{35} + 5\ q^{36} + 4\ q^{39} + 2\ q^{40} + 4\ q^{41} + 4\ q^{42} + 2\ q^{44} + 5\ q^{45} + 4\ q^{46} + 4\ q^{47} + 6\ q^{49} + 4\ q^{50} + 4\ q^{51} + 3\ q^{52} + 2\ q^{53} + 8\ q^{54} + 2\ q^{55} + 4\ q^{60} + 8\ q^{61} + 2\ q^{62} + 4\ q^{64} + 6\ q^{65} + 4\ q^{66} + q^{68} + 4\ q^{69} + 2\ q^{70} + 2\ q^{71} + 7\ q^{74} + 4\ q^{75} + 2\ q^{77} + 4\ q^{78} + 2\ q^{79} + 4\ q^{80} + 2\ q^{81} + 2\ q^{82} + q^{83} + 2\ q^{84} + 2\ q^{86} + 4\ q^{88} + 2\ q^{89} + 2\ q^{90} + 2\ q^{93} + 6\ q^{94} + 4\ q^{95} + 4\ q^{96} + 4\ q^{98} + 2\ q^{102} + 2\ q^{103} + 4\ q^{104} + 2\ q^{105} + 3\ q^{106} + 2\ q^{107} + 2\ q^{108} + 2\ q^{109} + 2\ q^{111} + 2\ q^{112} + 2\ q^{113} + 2\ q^{114} + 2\ q^{115} + 2\ q^{116} + 3\ q^{117} + 2\ q^{120} + 4\ q^{122} + 2\ q^{124} + q^{125} + 2\ q^{126} + 2\ q^{128} + 3\ q^{131} + q^{133} + 4\ q^{135} + 2\ q^{136} + 2\ q^{138} + 2\ q^{139} + 2\ q^{141} + 8\ q^{144} + 2\ q^{145} + 4\ q^{150} + 3\ q^{155} + 4\ q^{156} + 2\ q^{157} + 2\ q^{160} + q^{161} + 2\ q^{162} + 2\ q^{164} + 4\ q^{170} + 2\ q^{172} + 2\ q^{173} + 2\ q^{178} + 2\ q^{179} + q^{180} + 2\ q^{184} + 2\ q^{187} + 2\ q^{188} + 2\ q^{192} + 2\ q^{198} + 2\ q^{201} + 2\ q^{204} + q^{206} + 2\ q^{208} + 2\ q^{210} + 2\ q^{213} + 2\ q^{218} + 2\ q^{219} + 2\ q^{224} + 2\ q^{226} + q^{229} + q^{232} + 2\ q^{240} + 2\ q^{250} + 2\ q^{251} + 4\ q^{254} + 2\ q^{267} + q^{280} + 2\ q^{294} + 2\ q^{296} + 2\ q^{302} + q^{313} + 2\ q^{344} + 2\ q^{358} + q^{428}$

Description

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

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) ) - 1

Created

Feb 26, 2016 at 10:42 by Christian Stump

Updated

Nov 16, 2017 at 11:43 by Martin Rubey