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

Author

Gus Wiseman, May 31 2019

Keywords

• nonn
• more

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.

Examples

			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)
		

Crossrefs

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

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

Programs

Mathematica

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