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Identifier

St001659: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 1

[2] => 2

[1,1] => 2

[3] => 3

[2,1] => 1

[1,1,1] => 3

[4] => 4

[3,1] => 2

[2,2] => 2

[2,1,1] => 2

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

[5] => 5

[4,1] => 3

[3,2] => 4

[3,1,1] => 4

[2,2,1] => 4

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

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

[6] => 6

[5,1] => 4

[4,2] => 6

[4,1,1] => 6

[3,3] => 6

[3,2,1] => 1

[3,1,1,1] => 6

[2,2,2] => 6

[2,2,1,1] => 6

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

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

[7] => 7

[6,1] => 5

[5,2] => 8

[5,1,1] => 8

[4,3] => 9

[4,2,1] => 2

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

[3,3,1] => 2

[3,2,2] => 2

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

[3,1,1,1,1] => 8

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

[2,2,1,1,1] => 8

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

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

[8] => 8

[7,1] => 6

[6,2] => 10

[6,1,1] => 10

[5,3] => 12

[5,2,1] => 3

[5,1,1,1] => 12

[4,4] => 12

[4,3,1] => 4

[4,2,2] => 4

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

[4,1,1,1,1] => 12

[3,3,2] => 4

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

[3,2,2,1] => 4

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

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

[2,2,2,2] => 12

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

[2,2,1,1,1,1] => 10

[2,1,1,1,1,1,1] => 6

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

[9] => 9

[8,1] => 7

[7,2] => 12

[7,1,1] => 12

[6,3] => 15

[6,2,1] => 4

[6,1,1,1] => 15

[5,4] => 16

[5,3,1] => 6

[5,2,2] => 6

[5,2,1,1] => 6

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

[4,4,1] => 6

[4,3,2] => 8

[4,3,1,1] => 8

[4,2,2,1] => 8

[4,2,1,1,1] => 6

[4,1,1,1,1,1] => 15

[3,3,3] => 6

[3,3,2,1] => 8

[3,3,1,1,1] => 6

[3,2,2,2] => 6

[3,2,2,1,1] => 6

[3,2,1,1,1,1] => 4

[3,1,1,1,1,1,1] => 12

[2,2,2,2,1] => 16

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

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

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

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

[10] => 10

[9,1] => 8

[8,2] => 14

[8,1,1] => 14

>>> Load all 1212 entries. <<<

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$F_{1} = q$

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

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

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

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

$F_{6} = q + 2\ q^{4} + 8\ q^{6}$

$F_{7} = 4\ q^{2} + 2\ q^{5} + 2\ q^{7} + 4\ q^{8} + 3\ q^{9}$

$F_{8} = 2\ q^{3} + 6\ q^{4} + 2\ q^{6} + 2\ q^{8} + 4\ q^{10} + 6\ q^{12}$

$F_{9} = 2\ q^{4} + 9\ q^{6} + 2\ q^{7} + 4\ q^{8} + 2\ q^{9} + 4\ q^{12} + 4\ q^{15} + 3\ q^{16}$

$F_{10} = q + 2\ q^{5} + 8\ q^{8} + 3\ q^{9} + 2\ q^{10} + 12\ q^{12} + 4\ q^{14} + 4\ q^{18} + 6\ q^{20}$

$F_{11} = 5\ q^{2} + 2\ q^{6} + 2\ q^{9} + 6\ q^{10} + 2\ q^{11} + 6\ q^{12} + 10\ q^{16} + 12\ q^{18} + 4\ q^{21} + 4\ q^{24} + 3\ q^{25}$

$F_{12} = 2\ q^{3} + 10\ q^{4} + 2\ q^{7} + 2\ q^{10} + 8\ q^{12} + 4\ q^{15} + 3\ q^{16} + 4\ q^{18} + 6\ q^{20} + 22\ q^{24} + 4\ q^{27} + 4\ q^{28} + 6\ q^{30}$

$F_{13} = 2\ q^{4} + 12\ q^{6} + 12\ q^{8} + 2\ q^{11} + 2\ q^{13} + 6\ q^{14} + 4\ q^{18} + 10\ q^{20} + 6\ q^{24} + 4\ q^{27} + 12\ q^{30} + 10\ q^{32} + 4\ q^{35} + 15\ q^{36}$

$F_{14} = 2\ q^{5} + 8\ q^{8} + 5\ q^{9} + 26\ q^{12} + 2\ q^{14} + 11\ q^{16} + 4\ q^{21} + 4\ q^{22} + 4\ q^{24} + 3\ q^{25} + 6\ q^{28} + 4\ q^{30} + 16\ q^{36} + 16\ q^{40} + 6\ q^{42} + 6\ q^{45} + 12\ q^{48}$

$F_{15} = q + 2\ q^{6} + 10\ q^{10} + 6\ q^{12} + 2\ q^{13} + 2\ q^{15} + 12\ q^{16} + 24\ q^{18} + 28\ q^{24} + 4\ q^{28} + 6\ q^{30} + 6\ q^{32} + 4\ q^{33} + 4\ q^{40} + 12\ q^{42} + 4\ q^{45} + 12\ q^{48} + 3\ q^{49} + 6\ q^{50} + 6\ q^{54} + 18\ q^{60} + 4\ q^{64}$

$F_{16} = 6\ q^{2} + 2\ q^{7} + 2\ q^{11} + 8\ q^{12} + 2\ q^{14} + 4\ q^{15} + 5\ q^{16} + 18\ q^{20} + 27\ q^{24} + 4\ q^{26} + 8\ q^{27} + 12\ q^{32} + 4\ q^{35} + 43\ q^{36} + 4\ q^{44} + 12\ q^{48} + 4\ q^{50} + 4\ q^{54} + 14\ q^{56} + 12\ q^{60} + 6\ q^{63} + 12\ q^{72} + 6\ q^{75} + 12\ q^{80}$

$F_{17} = 2\ q^{3} + 15\ q^{4} + 2\ q^{8} + 2\ q^{12} + 8\ q^{14} + 2\ q^{15} + 2\ q^{17} + 4\ q^{18} + 6\ q^{20} + 6\ q^{22} + 12\ q^{24} + 4\ q^{28} + 22\ q^{30} + 9\ q^{32} + 16\ q^{36} + 4\ q^{39} + 18\ q^{40} + 6\ q^{42} + 40\ q^{48} + 32\ q^{54} + 4\ q^{55} + 4\ q^{60} + 4\ q^{63} + 11\ q^{64} + 8\ q^{70} + 12\ q^{72} + 12\ q^{84} + 12\ q^{90} + 6\ q^{96} + 12\ q^{100}$

Description

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

References

[1] wikipedia:Rook_polynomial

Code

def statistic(la):
    return next(v for v in reversed(matrix([[1]*p + [0]*(la[0]-p) for p in la]).rook_vector()) if v)

Created

Dec 22, 2020 at 13:32 by Martin Rubey

Updated

Dec 22, 2020 at 13:32 by Martin Rubey