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Identifier

St001177: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 0

[1,1] => 2

[3] => 0

[2,1] => 3

[1,1,1] => 6

[4] => 0

[3,1] => 4

[2,2] => 6

[2,1,1] => 8

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

[5] => 0

[4,1] => 5

[3,2] => 8

[3,1,1] => 10

[2,2,1] => 12

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

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

[6] => 0

[5,1] => 6

[4,2] => 10

[4,1,1] => 12

[3,3] => 12

[3,2,1] => 15

[3,1,1,1] => 18

[2,2,2] => 18

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

[2,1,1,1,1] => 24

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

[7] => 0

[6,1] => 7

[5,2] => 12

[5,1,1] => 14

[4,3] => 15

[4,2,1] => 18

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

[3,3,1] => 20

[3,2,2] => 22

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

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

[2,2,2,1] => 27

[2,2,1,1,1] => 30

[2,1,1,1,1,1] => 35

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

[8] => 0

[7,1] => 8

[6,2] => 14

[6,1,1] => 16

[5,3] => 18

[5,2,1] => 21

[5,1,1,1] => 24

[4,4] => 20

[4,3,1] => 24

[4,2,2] => 26

[4,2,1,1] => 28

[4,1,1,1,1] => 32

[3,3,2] => 28

[3,3,1,1] => 30

[3,2,2,1] => 32

[3,2,1,1,1] => 35

[3,1,1,1,1,1] => 40

[2,2,2,2] => 36

[2,2,2,1,1] => 38

[2,2,1,1,1,1] => 42

[2,1,1,1,1,1,1] => 48

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

[9] => 0

[8,1] => 9

[7,2] => 16

[7,1,1] => 18

[6,3] => 21

[6,2,1] => 24

[6,1,1,1] => 27

[5,4] => 24

[5,3,1] => 28

[5,2,2] => 30

[5,2,1,1] => 32

[5,1,1,1,1] => 36

[4,4,1] => 30

[4,3,2] => 33

[4,3,1,1] => 35

[4,2,2,1] => 37

[4,2,1,1,1] => 40

[4,1,1,1,1,1] => 45

[3,3,3] => 36

[3,3,2,1] => 39

[3,3,1,1,1] => 42

[3,2,2,2] => 42

[3,2,2,1,1] => 44

[3,2,1,1,1,1] => 48

[3,1,1,1,1,1,1] => 54

[2,2,2,2,1] => 48

[2,2,2,1,1,1] => 51

[2,2,1,1,1,1,1] => 56

[2,1,1,1,1,1,1,1] => 63

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

[10] => 0

[9,1] => 10

[8,2] => 18

[8,1,1] => 20

[7,3] => 24

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$F_{1} = 1$

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

$F_{3} = 1 + q^{3} + q^{6}$

$F_{4} = 1 + q^{4} + q^{6} + q^{8} + q^{12}$

$F_{5} = 1 + q^{5} + q^{8} + q^{10} + q^{12} + q^{15} + q^{20}$

$F_{6} = 1 + q^{6} + q^{10} + 2\ q^{12} + q^{15} + 2\ q^{18} + q^{20} + q^{24} + q^{30}$

$F_{7} = 1 + q^{7} + q^{12} + q^{14} + q^{15} + q^{18} + q^{20} + q^{21} + q^{22} + q^{24} + q^{27} + q^{28} + q^{30} + q^{35} + q^{42}$

$F_{8} = 1 + q^{8} + q^{14} + q^{16} + q^{18} + q^{20} + q^{21} + 2\ q^{24} + q^{26} + 2\ q^{28} + q^{30} + 2\ q^{32} + q^{35} + q^{36} + q^{38} + q^{40} + q^{42} + q^{48} + q^{56}$

$F_{9} = 1 + q^{9} + q^{16} + q^{18} + q^{21} + 2\ q^{24} + q^{27} + q^{28} + 2\ q^{30} + q^{32} + q^{33} + q^{35} + 2\ q^{36} + q^{37} + q^{39} + q^{40} + 2\ q^{42} + q^{44} + q^{45} + 2\ q^{48} + q^{51} + q^{54} + q^{56} + q^{63} + q^{72}$

$F_{10} = 1 + q^{10} + q^{18} + q^{20} + q^{24} + q^{27} + q^{28} + 2\ q^{30} + q^{32} + q^{34} + q^{35} + q^{36} + q^{38} + 3\ q^{40} + 3\ q^{42} + 2\ q^{45} + 3\ q^{48} + 3\ q^{50} + q^{52} + q^{54} + q^{55} + q^{56} + q^{58} + 2\ q^{60} + q^{62} + q^{63} + q^{66} + q^{70} + q^{72} + q^{80} + q^{90}$

$F_{11} = 1 + q^{11} + q^{20} + q^{22} + q^{27} + q^{30} + q^{32} + q^{33} + q^{35} + q^{36} + q^{38} + 2\ q^{40} + q^{42} + q^{43} + q^{44} + q^{45} + q^{46} + q^{47} + 2\ q^{48} + 2\ q^{50} + q^{51} + q^{53} + 2\ q^{54} + 2\ q^{55} + 2\ q^{56} + q^{57} + q^{59} + 2\ q^{60} + 2\ q^{62} + q^{63} + q^{64} + q^{65} + q^{66} + q^{67} + q^{68} + 2\ q^{70} + q^{72} + q^{74} + q^{75} + q^{77} + q^{78} + q^{80} + q^{83} + q^{88} + q^{90} + q^{99} + q^{110}$

$F_{12} = 1 + q^{12} + q^{22} + q^{24} + q^{30} + q^{33} + 2\ q^{36} + 2\ q^{40} + 2\ q^{42} + q^{44} + q^{45} + 3\ q^{48} + q^{50} + 2\ q^{52} + 3\ q^{54} + q^{55} + q^{56} + 2\ q^{57} + 5\ q^{60} + 2\ q^{62} + q^{63} + 2\ q^{64} + 3\ q^{66} + 2\ q^{68} + q^{69} + 2\ q^{70} + 5\ q^{72} + 2\ q^{75} + q^{76} + q^{77} + 3\ q^{78} + 2\ q^{80} + q^{82} + 3\ q^{84} + q^{87} + q^{88} + 2\ q^{90} + 2\ q^{92} + 2\ q^{96} + q^{99} + q^{102} + q^{108} + q^{110} + q^{120} + q^{132}$

$F_{13} = 1 + q^{13} + q^{24} + q^{26} + q^{33} + q^{36} + q^{39} + q^{40} + q^{44} + q^{45} + q^{46} + 2\ q^{48} + q^{50} + q^{52} + q^{53} + q^{54} + q^{55} + q^{56} + q^{57} + q^{58} + 3\ q^{60} + q^{61} + 2\ q^{63} + q^{64} + q^{65} + 3\ q^{66} + q^{67} + 2\ q^{68} + 2\ q^{69} + q^{70} + q^{71} + 3\ q^{72} + q^{73} + q^{74} + q^{75} + 3\ q^{76} + q^{77} + 3\ q^{78} + q^{79} + 3\ q^{80} + q^{81} + q^{82} + q^{83} + 3\ q^{84} + q^{85} + q^{86} + 2\ q^{87} + 2\ q^{88} + q^{89} + 3\ q^{90} + q^{91} + q^{92} + 2\ q^{93} + q^{95} + 3\ q^{96} + q^{98} + q^{99} + q^{100} + q^{101} + q^{102} + q^{103} + q^{104} + q^{106} + 2\ q^{108} + q^{110} + q^{111} + q^{112} + q^{116} + q^{117} + q^{120} + q^{123} + q^{130} + q^{132} + q^{143} + q^{156}$

$F_{14} = 1 + q^{14} + q^{26} + q^{28} + q^{36} + q^{39} + q^{42} + q^{44} + q^{48} + 2\ q^{50} + q^{52} + q^{54} + q^{55} + 2\ q^{56} + q^{58} + 2\ q^{60} + q^{62} + q^{63} + q^{64} + q^{65} + 2\ q^{66} + q^{68} + q^{69} + 3\ q^{70} + q^{71} + 3\ q^{72} + 3\ q^{74} + 2\ q^{76} + 2\ q^{77} + 3\ q^{78} + 3\ q^{80} + 2\ q^{82} + q^{83} + 6\ q^{84} + 3\ q^{86} + q^{87} + 2\ q^{88} + q^{89} + 3\ q^{90} + 3\ q^{91} + 3\ q^{92} + q^{93} + 2\ q^{94} + q^{95} + 3\ q^{96} + 6\ q^{98} + q^{99} + 2\ q^{100} + 3\ q^{102} + 3\ q^{104} + 2\ q^{105} + 2\ q^{106} + 3\ q^{108} + 3\ q^{110} + q^{111} + 3\ q^{112} + q^{113} + q^{114} + 2\ q^{116} + q^{117} + q^{118} + q^{119} + q^{120} + 2\ q^{122} + q^{124} + 2\ q^{126} + q^{127} + q^{128} + q^{130} + 2\ q^{132} + q^{134} + q^{138} + q^{140} + q^{143} + q^{146} + q^{154} + q^{156} + q^{168} + q^{182}$

$F_{15} = 1 + q^{15} + q^{28} + q^{30} + q^{39} + q^{42} + q^{45} + q^{48} + q^{52} + q^{54} + q^{55} + q^{56} + 3\ q^{60} + 2\ q^{63} + q^{65} + q^{66} + q^{67} + 3\ q^{70} + 3\ q^{72} + 3\ q^{75} + q^{77} + 4\ q^{78} + 2\ q^{80} + q^{81} + q^{82} + q^{83} + 4\ q^{84} + 2\ q^{85} + 3\ q^{87} + 2\ q^{88} + 5\ q^{90} + 2\ q^{91} + 2\ q^{92} + 3\ q^{93} + q^{94} + 2\ q^{95} + 3\ q^{96} + 3\ q^{98} + 3\ q^{99} + 4\ q^{100} + 5\ q^{102} + q^{103} + q^{104} + 6\ q^{105} + q^{106} + q^{107} + 5\ q^{108} + 4\ q^{110} + 3\ q^{111} + 3\ q^{112} + 3\ q^{114} + 2\ q^{115} + q^{116} + 3\ q^{117} + 2\ q^{118} + 2\ q^{119} + 5\ q^{120} + 2\ q^{122} + 3\ q^{123} + 2\ q^{125} + 4\ q^{126} + q^{127} + q^{128} + q^{129} + 2\ q^{130} + 4\ q^{132} + q^{133} + 3\ q^{135} + 3\ q^{138} + 3\ q^{140} + q^{143} + q^{144} + q^{145} + 2\ q^{147} + 3\ q^{150} + q^{154} + q^{155} + q^{156} + q^{158} + q^{162} + q^{165} + q^{168} + q^{171} + q^{180} + q^{182} + q^{195} + q^{210}$

$F_{16} = 1 + q^{16} + q^{30} + q^{32} + q^{42} + q^{45} + q^{48} + q^{52} + q^{56} + q^{58} + 2\ q^{60} + q^{64} + q^{65} + q^{66} + q^{68} + 2\ q^{70} + 3\ q^{72} + q^{75} + q^{76} + q^{77} + 2\ q^{78} + 2\ q^{80} + q^{81} + q^{82} + 3\ q^{84} + q^{85} + 2\ q^{86} + 3\ q^{88} + 4\ q^{90} + q^{91} + 2\ q^{92} + 2\ q^{93} + q^{94} + 5\ q^{96} + q^{97} + 3\ q^{98} + q^{99} + 5\ q^{100} + 4\ q^{102} + 4\ q^{104} + 3\ q^{105} + q^{106} + 2\ q^{107} + 5\ q^{108} + q^{109} + 4\ q^{110} + q^{111} + 7\ q^{112} + 5\ q^{114} + 2\ q^{115} + 2\ q^{116} + 2\ q^{117} + 3\ q^{118} + q^{119} + 9\ q^{120} + q^{121} + 3\ q^{122} + 2\ q^{123} + 2\ q^{124} + 2\ q^{125} + 5\ q^{126} + 7\ q^{128} + q^{129} + 4\ q^{130} + q^{131} + 5\ q^{132} + 2\ q^{133} + q^{134} + 3\ q^{135} + 4\ q^{136} + 4\ q^{138} + 5\ q^{140} + q^{141} + 3\ q^{142} + q^{143} + 5\ q^{144} + q^{146} + 2\ q^{147} + 2\ q^{148} + q^{149} + 4\ q^{150} + 3\ q^{152} + 2\ q^{154} + q^{155} + 3\ q^{156} + q^{158} + q^{159} + 2\ q^{160} + 2\ q^{162} + q^{163} + q^{164} + q^{165} + 3\ q^{168} + 2\ q^{170} + q^{172} + q^{174} + q^{175} + q^{176} + 2\ q^{180} + q^{182} + q^{184} + q^{188} + q^{192} + q^{195} + q^{198} + q^{208} + q^{210} + q^{224} + q^{240}$

Description

Twice the mean value of the major index among all standard Young tableaux of a partition.
For a partition

$\lambda$ of

$n$ , this mean value is given in [1, Proposition 3.1] by

$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$
where

$\lambda_i$ is the size of the

$i$ -th row of

$\lambda$ and

$\lambda_i'$ is the size of the

$i$ -th column.

References

[1] , Adin, R. M., Roichman, Y. Descent functions and random Young tableaux MathSciNet:1841639 arXiv:math/9910165

Code

def statistic(L):
    n = sum(L)
    return binomial(n,2) - sum(binomial(i,2) for i in L) + sum(binomial(i,2) for i in L.conjugate())

Created

May 07, 2018 at 20:47 by Christian Stump

Updated

May 07, 2018 at 20:47 by Christian Stump