A019472 Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

0, 0, 1, 7, 51, 421, 3963, 42253, 505515, 6724381, 98618763, 1582715773, 27612565995, 520631327581, 10554164679243, 228975516609853, 5294731892093355, 130015079601039901, 3379132289551117323, 92679942218919579133, 2675254894236207563115, 81073734056332364441821

Offset: 0

Author

Robert Ware (bware(AT)wam.umd.edu)

Keywords

• nonn
• easy
• nice

Comments

From Gus Wiseman, Jun 24 2020: (Start)

Equivalently, a(n) is number of (1,1)-matching sequences of length n that cover an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 7 sequences are:

(1,1) (1,1,1)

(1,1,2)

(1,2,1)

(1,2,2)

(2,1,1)

(2,1,2)

(2,2,1)

Missing from this list are:

(1,2) (1,2,3)

(2,1) (1,3,2)

(2,1,3)

(2,3,1)

(3,1,2)

(3,2,1)

(End)

Links

• Wikipedia, Weak ordering
• Wikipedia, Permutation pattern
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

Cf. A000670, A052875.

(1,1)-avoiding patterns are counted by A000142.

(1,2)-matching patterns are counted by A056823.

(1,1)-matching compositions are counted by A261982.

(1,1)-matching compositions are ranked by A335488.

Patterns matched by patterns are counted by A335517.

Cf. A056986, A333217, A335454, A335456, A335515.

Programs

Mathematica

a[n_] := Sum[(-1)^(j-i)*Binomial[j, i]*i^n, {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 26 2016, after Peter Luschny *)

Sage

def A019472(n):
    return add(add((-1)^(j-i)*binomial(j, i)*i^n for i in range(n)) for j in range(n))
[A019472(n) for n in range(21)] # Peter Luschny, Jul 22 2014

Formula

a(n) = A000670(n) - n!. - corrected by Eugene McDonnell, May 12 2000

a(n) = Sum_{j=0..n-1} Sum_{i=0..n-1} (-1)^(j-i)*C(j, i)*i^n. - Peter Luschny, Jul 22 2014