A325850 - OEIS

%I #7 Jun 02 2019 00:48:50

%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 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.

%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