- FindStat

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Identifier

St000421: Dyck paths ⟶ ℤ (values match St000418The number of Dyck paths that are weakly below a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + 3\ q^{3} + 2\ q^{4} + q^{7} + 2\ q^{9} + q^{12} + q^{13}$

$F_{5} = 1 + 4\ q + 6\ q^{3} + 3\ q^{4} + 4\ q^{7} + 6\ q^{9} + 2\ q^{12} + 2\ q^{13} + q^{15} + 3\ q^{19} + q^{24} + 2\ q^{25} + 2\ q^{27} + q^{33} + 2\ q^{36} + q^{40} + q^{41}$

$F_{6} = 1 + 5\ q + 10\ q^{3} + 4\ q^{4} + 10\ q^{7} + 12\ q^{9} + 3\ q^{12} + 3\ q^{13} + 5\ q^{15} + 12\ q^{19} + 3\ q^{24} + 6\ q^{25} + 6\ q^{27} + q^{31} + 2\ q^{33} + 4\ q^{36} + 4\ q^{39} + 2\ q^{40} + 2\ q^{41} + 3\ q^{49} + 3\ q^{51} + 3\ q^{55} + 2\ q^{64} + 2\ q^{67} + 2\ q^{69} + 4\ q^{73} + 2\ q^{81} + 2\ q^{83} + q^{88} + 2\ q^{96} + q^{97} + q^{105} + 2\ q^{108} + 2\ q^{111} + q^{121} + 2\ q^{125} + q^{130} + q^{131}$

$F_{7} = 1 + 6\ q + 15\ q^{3} + 5\ q^{4} + 20\ q^{7} + 20\ q^{9} + 4\ q^{12} + 4\ q^{13} + 15\ q^{15} + 30\ q^{19} + 6\ q^{24} + 12\ q^{25} + 12\ q^{27} + 6\ q^{31} + 3\ q^{33} + 6\ q^{36} + 20\ q^{39} + 3\ q^{40} + 3\ q^{41} + 12\ q^{49} + 12\ q^{51} + 12\ q^{55} + q^{63} + 6\ q^{64} + 6\ q^{67} + 6\ q^{69} + 12\ q^{73} + 5\ q^{79} + 6\ q^{81} + 6\ q^{83} + 2\ q^{88} + 4\ q^{96} + 2\ q^{97} + 6\ q^{99} + 4\ q^{103} + 2\ q^{105} + 4\ q^{108} + 8\ q^{111} + 2\ q^{121} + q^{124} + 4\ q^{125} + 6\ q^{129} + 2\ q^{130} + 2\ q^{131} + 3\ q^{135} + 6\ q^{139} + 6\ q^{147} + 3\ q^{163} + 3\ q^{167} + q^{168} + 2\ q^{169} + 2\ q^{177} + 2\ q^{181} + 4\ q^{184} + 4\ q^{193} + 3\ q^{195} + 2\ q^{204} + 2\ q^{209} + 2\ q^{211} + 4\ q^{217} + 4\ q^{223} + q^{232} + 2\ q^{243} + 4\ q^{251} + 2\ q^{253} + 2\ q^{256} + 2\ q^{261} + 2\ q^{263} + q^{276} + 2\ q^{280} + 2\ q^{285} + q^{289} + 2\ q^{293} + q^{298} + 2\ q^{312} + 2\ q^{321} + 2\ q^{324} + 2\ q^{335} + 2\ q^{336} + 2\ q^{350} + 2\ q^{353} + q^{364} + 2\ q^{377} + q^{380} + q^{391} + 2\ q^{397} + 2\ q^{401} + q^{416} + 2\ q^{421} + q^{427} + q^{428}$

Description

The number of Dyck paths that are weakly below a 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(D,A) ) - 1

Created

Feb 26, 2016 at 10:38 by Christian Stump

Updated

Feb 26, 2016 at 10:38 by Christian Stump