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A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

%I A321280 #25 Dec 16 2020 19:15:19
%S A321280 1,1,1,1,2,1,8,1,22,22,1,52,172,1,114,856,604,1,240,3488,7296,1,494,
%T A321280 12746,54746,31238,1,1004,43628,330068,518324,1,2026,143244,1756878,
%U A321280 5300418,2620708,1,4072,457536,8641800,43235304,55717312,1,8166,1434318,40298572,309074508,728888188,325024572
%N A321280 Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.
%H A321280 Alois P. Heinz, <a href="/A321280/b321280.txt">Rows n = 0..100, flattened</a>
%H A321280 S. Spiro, <a href="https://arxiv.org/abs/1810.00993">Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic</a>, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
%H A321280 David G. L. Wang, T. Zhao, <a href="https://arxiv.org/abs/2009.05973">The peak and descent statistics over ballot permutations</a>, arXiv:2009.05973 [math.CO], 2020.
%e A321280 Triangle T(n,k) begins:
%e A321280   1;
%e A321280   1;
%e A321280   1;
%e A321280   1,     2;
%e A321280   1,     8;
%e A321280   1,    22,      22;
%e A321280   1,    52,     172;
%e A321280   1,   114,     856,       604;
%e A321280   1,   240,    3488,      7296;
%e A321280   1,   494,   12746,     54746,      31238;
%e A321280   1,  1004,   43628,    330068,     518324;
%e A321280   1,  2026,  143244,   1756878,    5300418,    2620708;
%e A321280   1,  4072,  457536,   8641800,   43235304,   55717312;
%e A321280   1,  8166, 1434318,  40298572,  309074508,  728888188,  325024572;
%e A321280   1, 16356, 4438540, 180969752, 2026885824, 7589067592, 8460090160;
%e A321280   ...
%p A321280 b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, 1/x,
%p A321280        add(expand(x*b(u-j, o-1+j, c-1)), j=1..u)+
%p A321280        add(b(u+j-1, o-j, c+1), j=1..o)))
%p A321280     end:
%p A321280 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(`if`(n=0, 1, b(n, 0, 1))):
%p A321280 seq(T(n), n=0..14);
%t A321280 b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, 1/x, Sum[Expand[ x*b[u - j, o - 1 + j, c - 1]], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]];
%t A321280 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ If[n == 0, 1, b[n, 0, 1]]];
%t A321280 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 08 2018, after _Alois P. Heinz_ *)
%Y A321280 Columns k=0-3 give: A000012, A005803 (for n>0), A321268, A321269.
%Y A321280 Row sums give A000246.
%Y A321280 T(2n+1,n) gives A177042.
%Y A321280 T(2n+2,n) gives A303285(n+1).
%Y A321280 Cf. A262124, A262125.
%K A321280 nonn,tabf
%O A321280 0,5
%A A321280 _Alois P. Heinz_, Nov 01 2018