A325850 Number of permutations of {1..n} whose differences of all degrees are nonzero.

1, 1, 2, 4, 18, 72, 446, 2804, 21560, 184364, 1788514

Offset

0,3

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

Links

Table of n, a(n) for n=0..10.

Example

The a(1) = 1 through a(4) = 18 permutations:
  (1)  (12)  (132)  (1243)
       (21)  (213)  (1324)
             (231)  (1342)
             (312)  (1423)
                    (2134)
                    (2143)
                    (2314)
                    (2413)
                    (2431)
                    (3124)
                    (3142)
                    (3241)
                    (3412)
                    (3421)
                    (4132)
                    (4213)
                    (4231)
                    (4312)

Mathematica

Table[Length[Select[Permutations[Range[n]], !MemberQ[Union@@Table[Differences[#, i], {i, Length[#]}], 0]&]], {n, 0, 5}]

Crossrefs

Dominated by A295370, the case for only differences of degree 2.

Cf. A049988, A175342, A238423, A279945, A325545, A325851, A325852, A325874, A325875.

Sequence in context: A303352 A226011 A174085 * A052689 A139104 A014448

Adjacent sequences: A325847 A325848 A325849 * A325851 A325852 A325853

Keyword

nonn,more

Author

Gus Wiseman, May 31 2019

Status

approved