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A316293
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Number T(n,k) of permutations p of [n] such that k is the maximum of the partial sums of the signed up-down jump sequence of 0,p; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.
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6
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1, 1, 2, 1, 5, 8, 5, 1, 16, 50, 79, 69, 34, 9, 1, 65, 314, 872, 1539, 1823, 1494, 856, 339, 89, 14, 1, 326, 2142, 8799, 24818, 50561, 76944, 89546, 80938, 57284, 31771, 13707, 4520, 1103, 188, 20, 1, 1957, 16248, 89273, 355271, 1070455, 2514044, 4705648
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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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EXAMPLE
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Triangle T(n,k) begins:
: 1;
: 1;
: 2;
: 1, 5;
: 8, 16;
: 5, 50, 65;
: 1, 79, 314, 326;
: 69, 872, 2142, 1957;
: 34, 1539, 8799, 16248, 13700;
: 9, 1823, 24818, 89273, 137356, 109601;
: 1, 1494, 50561, 355271, 947713, 1287350, 986410;
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MAPLE
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b:= proc(u, o, c, k) option remember;
`if`(c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+j, k), j=1..u)+
add(b(u+j-1, o-j, c-j, k), j=1..o)))
end:
T:= (n, k)-> b(n, 0$2, k) -`if`(k=0, 0, b(n, 0$2, k-1)):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..8);
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MATHEMATICA
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b[u_, o_, c_, k_] := b[u, o, c, k] = If[c > k, 0, If[u + o == 0, 1,
Sum[b[u - j, o - 1 + j, c + j, k], {j, 1, u}] +
Sum[b[u + j - 1, o - j, c - j, k], {j, 1, o}]]];
T[n_, k_] := b[n, 0, 0, k] - If[k == 0, 0, b[n, 0, 0, k - 1]];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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