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%I #27 Jul 09 2018 04:29:34
%S 1,0,1,0,1,1,0,1,2,2,0,1,5,5,5,0,1,9,12,14,16,0,1,17,36,36,47,52,0,1,
%T 31,81,98,117,166,189,0,1,57,174,327,327,425,627,683,0,1,101,413,788,
%U 988,1116,1633,2400,2621,0,1,185,889,1890,3392,3392,4291,6471,9459,10061
%N Number T(n,k) of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with k; triangle T(n,k), n>=0, 0<=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="/A291684/b291684.txt">Rows n = 0..140, flattened</a>
%F Sum_{k=0..n} T(n,k) = T(n+1,n+1) = A291685(n).
%F T(2n,n) = T(2n,n+1) for all n>0.
%e T(3,1) = 1: 123.
%e T(3,2) = 2: 213, 231.
%e T(3,3) = 2: 312, 321.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 2;
%e 0, 1, 5, 5, 5;
%e 0, 1, 9, 12, 14, 16;
%e 0, 1, 17, 36, 36, 47, 52;
%e 0, 1, 31, 81, 98, 117, 166, 189;
%e 0, 1, 57, 174, 327, 327, 425, 627, 683;
%e 0, 1, 101, 413, 788, 988, 1116, 1633, 2400, 2621;
%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 T:= (n, k)-> b(0, n, k)-`if`(k=0, 0, b(0, n, k-1)):
%p seq(seq(T(n,k), k=0..n), n=0..12);
%t b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, j], {j, 1, Min[t, u]}] + Sum[b[u + j - 1, o - j, j], {j, 1, Min[t, o]}]];
%t T[n_, k_] := b[0, n, k] - If[k == 0, 0, b[0, n, k - 1]];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 09 2018, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A057427, A292168, A292169, A292170, A292171, A292172, A292173, A292174, A292175, A292176.
%Y Row sums and T(n+1,n+1) give A291685.
%Y T(2n,n) gives A291688, T(2n+1,n+1) gives A303203, T(n,ceiling(n/2)) gives A303204.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Aug 29 2017
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