A262124 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.

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, 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, 9, 45, 224, 1445, 6128, 7936, 0, 1, 1, 1, 3, 9, 45, 225, 1568, 9985, 55152, 50521, 0

Offset

0,14

Links

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Formula

A(n,k) = Sum_{i=0..k} A262125(n,i).

Example

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.
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.
A(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
A(4,2) = 8: 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
A(4,3) = 9: 1234, 1243, 1324, 1342, 1423, 2314, 2341, 2413, 3412.
Square array A(n,k) begins:
  1,   1,    1,    1,    1,    1,    1,    1, ...
  1,   1,    1,    1,    1,    1,    1,    1, ...
  0,   1,    1,    1,    1,    1,    1,    1, ...
  0,   2,    3,    3,    3,    3,    3,    3, ...
  0,   5,    8,    9,    9,    9,    9,    9, ...
  0,  16,   40,   44,   45,   45,   45,   45, ...
  0,  61,  162,  219,  224,  225,  225,  225, ...
  0, 272, 1134, 1445, 1568, 1574, 1575, 1575, ...

Maple

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)-> `if`(n=0, 1, (p-> add(coeff(p, x, i), i=0..min(n, k))
              )(add(b(j-1, n-j, 0), j=1..n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

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 *)

Crossrefs

Columns k=1-10 give: A000111, A262126, A262128, A262129, A262130, A262131, A262132, A262133, A262134, A262135.

Main diagonal gives A000246.

Cf. A258829, A262125.

Sequence in context: A187596 A263863 A134655 * A199954 A333580 A375924

Adjacent sequences: A262121 A262122 A262123 * A262125 A262126 A262127

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 11 2015

Status

approved