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%I #18 Oct 25 2018 15:07:23
%S 1,1,1,1,1,0,1,1,1,0,1,1,1,2,0,1,1,1,3,5,0,1,1,1,3,8,16,0,1,1,1,3,9,
%T 40,61,0,1,1,1,3,9,44,162,272,0,1,1,1,3,9,45,219,1134,1385,0,1,1,1,3,
%U 9,45,224,1445,6128,7936,0,1,1,1,3,9,45,225,1568,9985,55152,50521,0
%N Number A(n,k) of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A262124/b262124.txt">Antidiagonals n = 0..100, flattened</a>
%F A(n,k) = Sum_{i=0..k} A262125(n,i).
%e p = 1423 is counted by T(4,1) because the up-down signature of p = 1423 is 1,-1,1 with partial sums 1,0,1.
%e q = 1432 is not counted by any T(4,k) because the up-down signature of q = 1432 is 1,-1,-1 with partial sums 1,0,-1.
%e A(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
%e A(4,2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
%e A(4,3) = 9: 1234, 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 2, 3, 3, 3, 3, 3, 3, ...
%e 0, 5, 8, 9, 9, 9, 9, 9, ...
%e 0, 16, 40, 44, 45, 45, 45, 45, ...
%e 0, 61, 162, 219, 224, 225, 225, 225, ...
%e 0, 272, 1134, 1445, 1568, 1574, 1575, 1575, ...
%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 A:= (n,k)-> `if`(n=0, 1, (p-> add(coeff(p, x, i), i=0..min(n, k))
%p )(add(b(j-1, n-j, 0), j=1..n))):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[u_, o_, c_] := b[u, o, c] = If[c<0, 0, If[u+o == 0, x^c, Function[p, Sum[ Coefficient[p, x, i]*x^Max[i, c], {i, 0, Exponent[p, 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}]]]]; A[n_, k_] := If[n==0, 1, Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, k]}]][ Sum[b[j-1, n-j, 0], {j, 1, n}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A000111, A262126, A262128, A262129, A262130, A262131, A262132, A262133, A262134, A262135.
%Y Main diagonal gives A000246.
%Y Cf. A258829, A262125.
%K nonn,tabl
%O 0,14
%A _Alois P. Heinz_, Sep 11 2015
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