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A262163
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Number A(n,k) of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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12
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1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 4, 5, 0, 1, 1, 2, 5, 16, 16, 0, 1, 1, 2, 5, 19, 54, 61, 0, 1, 1, 2, 5, 20, 82, 324, 272, 0, 1, 1, 2, 5, 20, 86, 454, 1532, 1385, 0, 1, 1, 2, 5, 20, 87, 516, 2795, 12256, 7936, 0, 1, 1, 2, 5, 20, 87, 521, 3135, 20346, 74512, 50521, 0
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OFFSET
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0,13
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LINKS
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FORMULA
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A(n,k) = Sum_{i=0..k} A258829(n,i).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, ...
0, 2, 4, 5, 5, 5, 5, 5, ...
0, 5, 16, 19, 20, 20, 20, 20, ...
0, 16, 54, 82, 86, 87, 87, 87, ...
0, 61, 324, 454, 516, 521, 522, 522, ...
0, 272, 1532, 2795, 3135, 3264, 3270, 3271, ...
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MAPLE
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b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,
(p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
A:= (n, k)-> (p-> add(coeff(p, x, i), i=0..min(n, k)))(b(0, n, 0)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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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_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, k]}]][b[0, n, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000111 for n>0, A262164, A262165, A262166, A262167, A262168, A262169, A262170, A262171, A262172.
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KEYWORD
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AUTHOR
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STATUS
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approved
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