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Identifier

St001815: Posets ⟶ ℤ

Values

([],1) => 1

([],2) => 3

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

([],3) => 13

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

([(0,1),(0,2)],3) => 6

([(0,2),(2,1)],3) => 4

([(0,2),(1,2)],3) => 6

([],4) => 75

([(2,3)],4) => 44

([(1,2),(1,3)],4) => 32

([(0,1),(0,2),(0,3)],4) => 26

([(0,2),(0,3),(3,1)],4) => 16

([(0,1),(0,2),(1,3),(2,3)],4) => 12

([(1,2),(2,3)],4) => 20

([(0,3),(3,1),(3,2)],4) => 12

([(1,3),(2,3)],4) => 32

([(0,3),(1,3),(3,2)],4) => 12

([(0,3),(1,3),(2,3)],4) => 26

([(0,3),(1,2)],4) => 26

([(0,3),(1,2),(1,3)],4) => 22

([(0,2),(0,3),(1,2),(1,3)],4) => 18

([(0,3),(2,1),(3,2)],4) => 8

([(0,3),(1,2),(2,3)],4) => 16

([],5) => 541

([(3,4)],5) => 308

([(2,3),(2,4)],5) => 220

([(1,2),(1,3),(1,4)],5) => 176

([(0,1),(0,2),(0,3),(0,4)],5) => 150

([(0,2),(0,3),(0,4),(4,1)],5) => 88

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 64

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 52

([(1,3),(1,4),(4,2)],5) => 104

([(0,3),(0,4),(4,1),(4,2)],5) => 64

([(1,2),(1,3),(2,4),(3,4)],5) => 76

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 24

([(0,3),(0,4),(3,2),(4,1)],5) => 52

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 44

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 36

([(2,3),(3,4)],5) => 132

([(1,4),(4,2),(4,3)],5) => 76

([(0,4),(4,1),(4,2),(4,3)],5) => 52

([(2,4),(3,4)],5) => 220

([(1,4),(2,4),(4,3)],5) => 76

([(0,4),(1,4),(4,2),(4,3)],5) => 36

([(1,4),(2,4),(3,4)],5) => 176

([(0,4),(1,4),(2,4),(4,3)],5) => 52

([(0,4),(1,4),(2,4),(3,4)],5) => 150

([(0,4),(1,4),(2,3)],5) => 126

([(0,4),(1,3),(2,3),(2,4)],5) => 102

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 90

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 78

([(0,4),(1,4),(2,3),(4,2)],5) => 24

([(0,4),(1,3),(2,3),(3,4)],5) => 64

([(0,4),(1,4),(2,3),(2,4)],5) => 114

([(0,4),(1,4),(2,3),(3,4)],5) => 88

([(1,4),(2,3)],5) => 176

([(1,4),(2,3),(2,4)],5) => 148

([(0,4),(1,2),(1,4),(2,3)],5) => 68

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 44

([(1,3),(1,4),(2,3),(2,4)],5) => 120

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 48

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 36

([(0,4),(1,2),(1,4),(4,3)],5) => 56

([(0,4),(1,2),(1,3)],5) => 126

([(0,4),(1,2),(1,3),(1,4)],5) => 114

([(0,2),(0,4),(3,1),(4,3)],5) => 40

([(0,4),(1,2),(1,3),(3,4)],5) => 80

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 32

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 64

([(0,3),(0,4),(1,2),(1,4)],5) => 102

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 90

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 78

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 60

([(0,3),(1,2),(1,4),(3,4)],5) => 68

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 48

([(1,4),(3,2),(4,3)],5) => 48

([(0,3),(3,4),(4,1),(4,2)],5) => 24

([(1,4),(2,3),(3,4)],5) => 104

([(0,4),(1,2),(2,4),(4,3)],5) => 32

([(0,3),(1,4),(4,2)],5) => 76

([(0,4),(3,2),(4,1),(4,3)],5) => 32

([(0,4),(1,2),(2,3),(2,4)],5) => 56

([(0,4),(2,3),(3,1),(4,2)],5) => 16

([(0,3),(1,2),(2,4),(3,4)],5) => 52

([(0,4),(1,2),(2,3),(3,4)],5) => 40

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 24

([],6) => 4683

([(4,5)],6) => 2612

([(3,4),(3,5)],6) => 1844

([(2,3),(2,4),(2,5)],6) => 1460

([(1,2),(1,3),(1,4),(1,5)],6) => 1232

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 1082

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 616

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 440

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 352

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 300

([(1,3),(1,4),(1,5),(5,2)],6) => 704

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 440

([(1,2),(1,3),(1,4),(3,5),(4,5)],6) => 504

([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => 404

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Description

The number of order preserving surjections from a poset to a total order.

Code

def statistic(P):
    p = P.order_polynomial()
    n = len(P)
    s = [(-1)^m*sum((-1)^k*binomial(m, k)*p.subs(q=k) for k in range(m+1)) for m in range(n+1)]
    return sum(s)

Created

Jul 08, 2022 at 20:41 by Martin Rubey

Updated

Jul 08, 2022 at 20:41 by Martin Rubey