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Identifier

St001389: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 2

[1,1] => 1

[3] => 3

[2,1] => 2

[1,1,1] => 1

[4] => 4

[3,1] => 3

[2,2] => 3

[2,1,1] => 2

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

[5] => 5

[4,1] => 4

[3,2] => 5

[3,1,1] => 3

[2,2,1] => 3

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

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

[6] => 6

[5,1] => 5

[4,2] => 7

[4,1,1] => 4

[3,3] => 6

[3,2,1] => 5

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

[2,2,2] => 4

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

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

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

[7] => 7

[6,1] => 6

[5,2] => 9

[5,1,1] => 5

[4,3] => 9

[4,2,1] => 7

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

[3,3,1] => 6

[3,2,2] => 7

[3,2,1,1] => 5

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

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

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

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

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

[8] => 8

[7,1] => 7

[6,2] => 11

[6,1,1] => 6

[5,3] => 12

[5,2,1] => 9

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

[4,4] => 10

[4,3,1] => 9

[4,2,2] => 10

[4,2,1,1] => 7

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

[3,3,2] => 9

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

[3,2,2,1] => 7

[3,2,1,1,1] => 5

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

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

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

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

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

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

[9] => 9

[8,1] => 8

[7,2] => 13

[7,1,1] => 7

[6,3] => 15

[6,2,1] => 11

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

[5,4] => 14

[5,3,1] => 12

[5,2,2] => 13

[5,2,1,1] => 9

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

[4,4,1] => 10

[4,3,2] => 14

[4,3,1,1] => 9

[4,2,2,1] => 10

[4,2,1,1,1] => 7

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

[3,3,3] => 10

[3,3,2,1] => 9

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

[3,2,2,2] => 9

[3,2,2,1,1] => 7

[3,2,1,1,1,1] => 5

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

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

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

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

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

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

[10] => 10

[9,1] => 9

[8,2] => 15

[8,1,1] => 8

[7,3] => 18

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$F_{1} = q$

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

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

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

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

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

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

$F_{8} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 2\ q^{6} + 3\ q^{7} + q^{8} + 3\ q^{9} + 2\ q^{10} + q^{11} + q^{12}$

$F_{9} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 2\ q^{6} + 3\ q^{7} + q^{8} + 5\ q^{9} + 3\ q^{10} + q^{11} + q^{12} + 2\ q^{13} + 2\ q^{14} + q^{15}$

$F_{10} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + q^{8} + 5\ q^{9} + 4\ q^{10} + q^{11} + 2\ q^{12} + 3\ q^{13} + 2\ q^{14} + 3\ q^{15} + 3\ q^{16} + 2\ q^{18} + q^{19}$

$F_{11} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + q^{8} + 5\ q^{9} + 4\ q^{10} + 3\ q^{11} + 2\ q^{12} + 3\ q^{13} + 3\ q^{14} + 3\ q^{15} + 3\ q^{16} + 2\ q^{17} + 2\ q^{18} + 4\ q^{19} + q^{20} + q^{21} + 2\ q^{22} + q^{23} + q^{24}$

$F_{12} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + q^{8} + 5\ q^{9} + 4\ q^{10} + 3\ q^{11} + 3\ q^{12} + 3\ q^{13} + 3\ q^{14} + 5\ q^{15} + 4\ q^{16} + 2\ q^{17} + 2\ q^{18} + 5\ q^{19} + 2\ q^{20} + 3\ q^{21} + 4\ q^{22} + 2\ q^{23} + 2\ q^{24} + 2\ q^{25} + 2\ q^{26} + 2\ q^{28} + q^{29} + q^{30}$

$F_{13} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + q^{8} + 5\ q^{9} + 4\ q^{10} + 3\ q^{11} + 3\ q^{12} + 5\ q^{13} + 3\ q^{14} + 5\ q^{15} + 4\ q^{16} + 2\ q^{17} + 3\ q^{18} + 5\ q^{19} + 2\ q^{20} + 5\ q^{21} + 4\ q^{22} + 2\ q^{23} + 3\ q^{24} + 5\ q^{25} + 2\ q^{26} + 2\ q^{27} + 3\ q^{28} + q^{29} + 4\ q^{30} + q^{31} + 2\ q^{32} + q^{33} + 3\ q^{34} + 2\ q^{37}$

$F_{14} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + 2\ q^{8} + 5\ q^{9} + 4\ q^{10} + 3\ q^{11} + 3\ q^{12} + 5\ q^{13} + 4\ q^{14} + 5\ q^{15} + 4\ q^{16} + 2\ q^{17} + 4\ q^{18} + 6\ q^{19} + 3\ q^{20} + 5\ q^{21} + 4\ q^{22} + 3\ q^{23} + 3\ q^{24} + 5\ q^{25} + 3\ q^{26} + 2\ q^{27} + 6\ q^{28} + 2\ q^{29} + 7\ q^{30} + 2\ q^{31} + 2\ q^{32} + 3\ q^{33} + 5\ q^{34} + 3\ q^{35} + q^{36} + 2\ q^{37} + q^{39} + 3\ q^{40} + q^{41} + 2\ q^{42} + 2\ q^{43} + q^{44} + q^{46}$

$F_{15} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + 2\ q^{8} + 5\ q^{9} + 4\ q^{10} + 3\ q^{11} + 3\ q^{12} + 5\ q^{13} + 4\ q^{14} + 7\ q^{15} + 4\ q^{16} + 2\ q^{17} + 4\ q^{18} + 6\ q^{19} + 3\ q^{20} + 6\ q^{21} + 5\ q^{22} + 3\ q^{23} + 3\ q^{24} + 7\ q^{25} + 3\ q^{26} + 2\ q^{27} + 6\ q^{28} + 3\ q^{29} + 7\ q^{30} + 4\ q^{31} + 2\ q^{32} + 5\ q^{33} + 7\ q^{34} + 5\ q^{35} + q^{36} + 3\ q^{37} + q^{38} + 2\ q^{39} + 4\ q^{40} + 2\ q^{41} + 4\ q^{42} + 2\ q^{43} + 2\ q^{44} + q^{45} + 4\ q^{46} + 4\ q^{47} + 2\ q^{48} + 2\ q^{50} + q^{51} + 2\ q^{52} + 2\ q^{55} + q^{56}$

$F_{16} = q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 4\ q^{7} + 2\ q^{8} + 6\ q^{9} + 4\ q^{10} + 3\ q^{11} + 3\ q^{12} + 5\ q^{13} + 4\ q^{14} + 7\ q^{15} + 5\ q^{16} + 2\ q^{17} + 4\ q^{18} + 6\ q^{19} + 3\ q^{20} + 7\ q^{21} + 6\ q^{22} + 3\ q^{23} + 3\ q^{24} + 8\ q^{25} + 3\ q^{26} + 3\ q^{27} + 6\ q^{28} + 3\ q^{29} + 7\ q^{30} + 5\ q^{31} + 2\ q^{32} + 5\ q^{33} + 9\ q^{34} + 6\ q^{35} + 5\ q^{36} + 5\ q^{37} + q^{38} + 2\ q^{39} + 6\ q^{40} + 2\ q^{41} + 6\ q^{42} + 3\ q^{43} + 2\ q^{44} + 4\ q^{45} + 4\ q^{46} + 4\ q^{47} + 3\ q^{48} + 2\ q^{49} + 3\ q^{50} + 3\ q^{51} + 4\ q^{52} + 3\ q^{53} + 3\ q^{54} + 3\ q^{55} + 2\ q^{56} + 2\ q^{58} + q^{59} + q^{60} + 3\ q^{61} + 3\ q^{62} + q^{63} + q^{64} + q^{66} + q^{67} + q^{70}$

Description

The number of partitions of the same length below the given integer partition.
For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Code

def statistic(la):
    k = len(la)
    return det(matrix([[binomial(la[k-i], j-i+1) for j in range(1, k+1)] for i in range(1, k+1)]))

Created

May 11, 2019 at 18:25 by Martin Rubey

Updated

May 11, 2019 at 18:25 by Martin Rubey