%I
%S 1,1,2,4,18,72,446,2804,21560,184364,1788514
%N Number of permutations of {1..n} whose differences of all degrees are nonzero.
%C 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 kth differences for k > 0 are the differences of the (k1)th differences. The differences of all degrees of a sequence are the union of its zeroth through mth differences, where m is the length of the sequence.
%e The a(1) = 1 through a(4) = 18 permutations:
%e (1) (12) (132) (1243)
%e (21) (213) (1324)
%e (231) (1342)
%e (312) (1423)
%e (2134)
%e (2143)
%e (2314)
%e (2413)
%e (2431)
%e (3124)
%e (3142)
%e (3241)
%e (3412)
%e (3421)
%e (4132)
%e (4213)
%e (4231)
%e (4312)
%t Table[Length[Select[Permutations[Range[n]],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,5}]
%Y Dominated by A295370, the case for only differences of degree 2.
%Y Cf. A049988, A175342, A238423, A279945, A325545, A325851, A325852, A325874, A325875.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, May 31 2019
