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A291688
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Number of permutations p of [2n] such that 0p has a nonincreasing jump sequence beginning with n.
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3
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1, 1, 5, 36, 327, 3392, 38795, 469662, 5935728, 77416352, 1035050705, 14094000938, 195075365778, 2734475097609, 38747262233793, 554199475506095, 7990492729051526, 115995691148658656, 1694340616136589743, 24882428969673439384, 367160435328847044586
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 5: 2134, 2314, 2341, 2413, 2431.
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, j), j=1..min(t, u))+
add(b(u+j-1, o-j, j), j=1..min(t, o)))
end:
a:= n-> b(0, 2*n, n)-`if`(n=0, 0, b(0, 2*n, n-1)):
seq(a(n), n=0..25);
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, j], {j, Min[t, u]}] +
Sum[b[u + j - 1, o - j, j], {j, Min[t, o]}]];
a[n_] := b[0, 2n, n] - If[n == 0, 0, b[0, 2n, n - 1]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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