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A316293 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.

%I A316293 #16 Mar 14 2021 11:42:56
%S A316293 1,1,2,1,5,8,5,1,16,50,79,69,34,9,1,65,314,872,1539,1823,1494,856,339,
%T A316293 89,14,1,326,2142,8799,24818,50561,76944,89546,80938,57284,31771,
%U A316293 13707,4520,1103,188,20,1,1957,16248,89273,355271,1070455,2514044,4705648
%N A316293 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.
%C A316293 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 A316293 Alois P. Heinz, <a href="/A316293/b316293.txt">Columns k = 0..40, flattened</a>
%e A316293 Triangle T(n,k) begins:
%e A316293 : 1;
%e A316293 :    1;
%e A316293 :       2;
%e A316293 :       1, 5;
%e A316293 :          8, 16;
%e A316293 :          5, 50,   65;
%e A316293 :          1, 79,  314,   326;
%e A316293 :             69,  872,  2142,   1957;
%e A316293 :             34, 1539,  8799,  16248,  13700;
%e A316293 :              9, 1823, 24818,  89273, 137356,  109601;
%e A316293 :              1, 1494, 50561, 355271, 947713, 1287350, 986410;
%p A316293 b:= proc(u, o, c, k) option remember;
%p A316293       `if`(c>k, 0, `if`(u+o=0, 1,
%p A316293        add(b(u-j, o-1+j, c+j, k), j=1..u)+
%p A316293        add(b(u+j-1, o-j, c-j, k), j=1..o)))
%p A316293     end:
%p A316293 T:= (n, k)-> b(n, 0$2, k) -`if`(k=0, 0, b(n, 0$2, k-1)):
%p A316293 seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..8);
%t A316293 b[u_, o_, c_, k_] := b[u, o, c, k] = If[c > k, 0, If[u + o == 0, 1,
%t A316293      Sum[b[u - j, o - 1 + j, c + j, k], {j, 1, u}] +
%t A316293      Sum[b[u + j - 1, o - j, c - j, k], {j, 1, o}]]];
%t A316293 T[n_, k_] := b[n, 0, 0, k] - If[k == 0, 0, b[n, 0, 0, k - 1]];
%t A316293 Table[Table[T[n, k], {n, k, k(k+1)/2}], {k, 0, 8}] // Flatten (* _Jean-François Alcover_, Mar 14 2021, after _Alois P. Heinz_ *)
%Y A316293 Row sums give A000142.
%Y A316293 Column sums give A316294.
%Y A316293 Main diagonal gives A000522.
%Y A316293 Cf. A000217, A258829, A291722, A303697, A316292 (same read by rows).
%K A316293 nonn,tabf
%O A316293 0,3
%A A316293 _Alois P. Heinz_, Jun 28 2018