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%I #21 Feb 27 2021 11:18:25
%S 1,1,2,1,5,8,16,5,50,65,1,79,314,326,69,872,2142,1957,34,1539,8799,
%T 16248,13700,9,1823,24818,89273,137356,109601,1,1494,50561,355271,
%U 947713,1287350,986410,856,76944,1070455,4923428,10699558,13281458,9864101
%N 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), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows.
%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="/A316292/b316292.txt">Rows n = 0..120, flattened</a>
%e Triangle T(n,k) begins:
%e : 1;
%e : 1;
%e : 2;
%e : 1, 5;
%e : 8, 16;
%e : 5, 50, 65;
%e : 1, 79, 314, 326;
%e : 69, 872, 2142, 1957;
%e : 34, 1539, 8799, 16248, 13700;
%e : 9, 1823, 24818, 89273, 137356, 109601;
%e : 1, 1494, 50561, 355271, 947713, 1287350, 986410;
%p b:= proc(u, o, c, k) option remember;
%p `if`(c>k, 0, `if`(u+o=0, 1,
%p add(b(u-j, o-1+j, c+j, k), j=1..u)+
%p add(b(u+j-1, o-j, c-j, k), j=1..o)))
%p end:
%p T:= (n, k)-> b(n, 0$2, k) -`if`(k=0, 0, b(n, 0$2, k-1)):
%p seq(seq(T(n, k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);
%t b[u_, o_, c_, k_] := b[u, o, c, k] =
%t If[c > k, 0, If[u + o == 0, 1,
%t Sum[b[u - j, o - 1 + j, c + j, k], {j, 1, u}] +
%t Sum[b[u + j - 1, o - j, c - j, k], {j, 1, o}]]];
%t T[n_, k_] := b[n, 0, 0, k] - If[k == 0, 0, b[n, 0, 0, k - 1]];
%t Table[Table[T[n, k], {k, Ceiling[(Sqrt[8n+1]-1)/2], n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 27 2021, after _Alois P. Heinz_ *)
%Y Row sums give A000142.
%Y Column sums give A316294.
%Y Main diagonal gives A000522.
%Y Cf. A002024, A123578, A258829, A291722, A303697, A316293 (same read by columns).
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Jun 28 2018
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