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%I #18 Jan 19 2020 07:29:24
%S 1,1,0,0,1,0,0,2,1,0,0,5,3,1,0,0,16,24,4,1,0,0,61,101,57,5,1,0,0,272,
%T 862,311,123,6,1,0,0,1385,4743,3857,778,254,7,1,0,0,7936,47216,27589,
%U 14126,1835,514,8,1,0,0,50521,322039,355751,111811,47673,4189,1031,9,1,0
%N Number T(n,k) of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A262125/b262125.txt">Rows n = 0..100, flattened</a>
%F T(n,k) = A262124(n,k) - A262124(n,k-1) for k>0, T(n,0) = A262124(n,0).
%e T(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
%e T(4,2) = 3: 1243, 1342, 2341.
%e T(4,3) = 1: 1234.
%e Triangle T(n,k) begins:
%e 1;
%e 1, 0;
%e 0, 1, 0;
%e 0, 2, 1, 0;
%e 0, 5, 3, 1, 0;
%e 0, 16, 24, 4, 1, 0;
%e 0, 61, 101, 57, 5, 1, 0;
%e 0, 272, 862, 311, 123, 6, 1, 0;
%e 0, 1385, 4743, 3857, 778, 254, 7, 1, 0;
%p b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,
%p (p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(
%p b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
%p end:
%p T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, i), i=0..n)
%p )(add(b(j-1, n-j, 0), j=1..n))):
%p seq(T(n), n=0..10);
%t b[u_, o_, c_] := b[u, o, c] = If[c<0, 0, If[u+o==0, x^c, Sum[Coefficient[ #, x, i]*x^Max[i, c], {i, 0, Exponent[#, x]}]]& @ Sum[b[u-j, o-1+j, c-1], {j, 1, u}] + Sum[b[u+j-1, o-j, c+1], {j, 1, o}]];
%t T[n_] := If[n==0, {1}, Table[Coefficient[#, x, i], {i, 0, n}]]& @ Sum[b[j-1, n-j, 0], {j, 1, n}];
%t T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, Jan 19 2020, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A000111 (for n>1), A320976, A320977, A320978, A320979, A320980, A320981, A320982, A320983, A320984.
%Y Row sums give A000246.
%Y T(2n,n) gives A262127.
%Y Cf. A258829, A262124.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 11 2015
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