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%I #6 Sep 10 2017 15:24:46
%S 5,14,36,98,327,788,1890,4523,11483,27209,61983,139568,323531,730494,
%T 1618531,3543509,7851556,17204960,37331967,80312122,173234350,
%U 372183777,794587802,1686743710,3579347697,7578433425,15984149737,33580700124,70467297970,147600492302
%N Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with four.
%C 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 Alois P. Heinz, <a href="/A292170/b292170.txt">Table of n, a(n) for n = 4..3365</a>
%e a(4) = 5: 4123, 4132, 4213, 4231, 4321.
%e a(5) = 14: 41235, 41325, 41352, 41523, 41532, 42135, 42315, 42351, 42513, 42531, 43215, 43251, 43521, 45321.
%e a(6) = 36: 412356, 413256, 413526, 413562, 413625, 413652, 415236, 415263, 415326, 415362, 415632, 421356, 423156, 423516, 423561, 425136, 425163, 425316, 425361, 425631, 432156, 432516, 432561, 435216, 435261, 435621, 453216, 453261, 453621, 456321, 463125, 463152, 463215, 463251, 463521, 465321.
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, j), j=1..min(t, u))+
%p add(b(u+j-1, o-j, j), j=1..min(t, o)))
%p end:
%p a:= n-> b(0, n, 4)-b(0, n, 3):
%p seq(a(n), n=4..50);
%Y Column k=4 of A291684.
%K nonn
%O 4,1
%A _Alois P. Heinz_, Sep 10 2017
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