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A291685 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.

%I A291685 #17 Aug 30 2021 04:06:43
%S A291685 1,1,2,5,16,52,189,683,2621,10061,40031,159201,650880,2657089,
%T A291685 11062682,46065143,194595138,822215099,3513875245,15021070567,
%U A291685 64785349064,279575206629,1214958544538,5283266426743,23106210465665,101120747493793,444614706427665
%N A291685 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.
%C A291685 An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
%H A291685 Alois P. Heinz, <a href="/A291685/b291685.txt">Table of n, a(n) for n = 0..500</a>
%e A291685 a(3) = 5 = 6 - 1 counts all permutations of {1,2,3} except 132 with jump sequence 1, 2, 1.
%p A291685 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A291685       add(b(u-j, o+j-1, j), j=1..min(t, u))+
%p A291685       add(b(u+j-1, o-j, j), j=1..min(t, o)))
%p A291685     end:
%p A291685 a:= n-> b(0, n$2):
%p A291685 seq(a(n), n=0..30);
%t A291685 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
%t A291685      Sum[b[u-j, o+j-1, j], {j, Min[t, u]}]+
%t A291685      Sum[b[u+j-1, o-j, j], {j, Min[t, o]}]];
%t A291685 a[n_] := b[0, n, n];
%t A291685 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A291685 Row sums and main diagonal (shifted) of A291684.
%Y A291685 Cf. A288910, A288911, A288912.
%K A291685 nonn
%O A291685 0,3
%A A291685 _Alois P. Heinz_, Aug 29 2017