A238354 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n.

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 1, 0, 0, 5, 1, 0, 1, 0, 0, 8, 1, 1, 0, 1, 0, 0, 11, 2, 0, 1, 0, 1, 0, 0, 17, 2, 1, 0, 1, 0, 1, 0, 0, 23, 3, 1, 1, 0, 1, 0, 1, 0, 0, 33, 4, 2, 0, 1, 0, 1, 0, 1, 0, 0, 45, 5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 63, 6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 84, 8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Keywords

• nonn
• tabl

Comments

Column k=0: T(n,0) = 1 + A047967(n).

Column k=1 is A238708.

Row sums are A000041.

Examples

			Triangle starts:
  00:    1;
  01:    1,  0;
  02:    2,  0, 0;
  03:    2,  1, 0, 0;
  04:    4,  0, 1, 0, 0;
  05:    5,  1, 0, 1, 0, 0;
  06:    8,  1, 1, 0, 1, 0, 0;
  07:   11,  2, 0, 1, 0, 1, 0, 0;
  08:   17,  2, 1, 0, 1, 0, 1, 0, 0;
  09:   23,  3, 1, 1, 0, 1, 0, 1, 0, 0;
  10:   33,  4, 2, 0, 1, 0, 1, 0, 1, 0, 0;
  11:   45,  5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  12:   63,  6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  13:   84,  8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  14:  114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  15:  150, 13, 4, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  ...
The 11 partitions of 6 together with their minimal ascents are:
  01:  [ 1 1 1 1 1 1 ]   0
  02:  [ 1 1 1 1 2 ]     0
  03:  [ 1 1 1 3 ]       0
  04:  [ 1 1 2 2 ]       0
  05:  [ 1 1 4 ]         0
  06:  [ 1 2 3 ]         1
  07:  [ 1 5 ]           4
  08:  [ 2 2 2 ]         0
  09:  [ 2 4 ]           2
  10:  [ 3 3 ]           0
  11:  [ 6 ]             0
There are 8 partitions of 6 with min ascent 0, 1 with min ascents 1, 2, and 4, giving row 6 of the triangle: 8, 1, 1, 0, 1, 0, 0.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A238353 (partitions by maximal ascent).

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1/x, `if`(i<1, 0,
      b(n, i-1, t)+`if`(i>n, 0, (p->`if`(t=0, p, add(coeff(
       p, x, j)*x^`if`(j<0, t-i, min(j, t-i)),
       j=-1..degree(p))))(b(n-i, i, i)))))
    end:
T:= n->(p->seq(coeff(p, x, k)+`if`(k=0, 1, 0), k=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1/x, If[i<1, 0, b[n, i-1, t]+If[i>n, 0, Function[{p}, If[t == 0, p, Sum[Coefficient[p, x, j]*x^If[j<0, t-i, Min[j, t-i]], {j, -1, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, k]+If[k == 0, 1, 0], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)