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A303204 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).

%I A303204 #12 Aug 30 2021 07:55:17
%S A303204 1,1,1,2,5,12,36,98,327,988,3392,10872,38795,129520,469662,1609176,
%T A303204 5935728,20786804,77416352,274792342,1035050705,3719296036,
%U A303204 14094000938,51119572738,195075365778,712918642042,2734475097609,10055531355652,38747262233793
%N A303204 Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).
%C A303204 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 A303204 Alois P. Heinz, <a href="/A303204/b303204.txt">Table of n, a(n) for n = 0..500</a>
%F A303204 a(n) = A291684(n,ceiling(n/2)).
%p A303204 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A303204       add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
%p A303204       add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
%p A303204     end:
%p A303204 a:= n-> `if`(n=0, 1, (j-> b(0, n, j)-b(0, n, j-1))(ceil(n/2))):
%p A303204 seq(a(n), n=0..30);
%t A303204 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
%t A303204      Sum[b[Sequence @@ Sort[{u-j, o+j-1}], j], {j, Min[t, u]}]+
%t A303204      Sum[b[Sequence @@ Sort[{u+j-1, o-j}], j], {j, Min[t, o]}]];
%t A303204 a[n_] := If[n == 0, 1,
%t A303204      Function[j, b[0, n, j] - b[0, n, j-1]][Ceiling[n/2]]];
%t A303204 Table[a[n], {n, 0, 30}]; (* _Jean-François Alcover_, Aug 30 2021, after _Alois P. Heinz_ *)
%Y A303204 Bisections give A291688 (even part), A303203 (odd part).
%Y A303204 Cf. A291684.
%K A303204 nonn
%O A303204 0,4
%A A303204 _Alois P. Heinz_, Apr 19 2018