A238353 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0, 2, 3, 1, 1, 0, 0, 4, 3, 2, 1, 1, 0, 0, 2, 6, 3, 2, 1, 1, 0, 0, 4, 6, 6, 2, 2, 1, 1, 0, 0, 3, 10, 6, 5, 2, 2, 1, 1, 0, 0, 4, 11, 11, 6, 4, 2, 2, 1, 1, 0, 0, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 0, 0, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 0, 0, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1, 0, 0

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Keywords

• nonn
• tabl

Comments

Reversed rows and also the columns converge to A002865 (setting A002865(0)=0).

Column k=0 is A000005 (n>=1), column k=1 is A237665.

Row sums are A000041.

Sum_{i=0..k} T(n,i) for k=0-9 gives: A000005, A034296, A224956, A238863, A238864, A238865, A238866, A238867, A238868, A238869.

Examples

			Triangle starts:
00:  1;
01:  1,  0;
02:  2,  0,  0;
03:  2,  1,  0,  0;
04:  3,  1,  1,  0,  0;
05:  2,  3,  1,  1,  0,  0;
06:  4,  3,  2,  1,  1,  0, 0;
07:  2,  6,  3,  2,  1,  1, 0, 0;
08:  4,  6,  6,  2,  2,  1, 1, 0, 0;
09:  3, 10,  6,  5,  2,  2, 1, 1, 0, 0;
10:  4, 11, 11,  6,  4,  2, 2, 1, 1, 0, 0;
11:  2, 16, 13, 10,  5,  4, 2, 2, 1, 1, 0, 0;
12:  6, 17, 19, 12,  9,  4, 4, 2, 2, 1, 1, 0, 0;
13:  2, 24, 24, 18, 11,  8, 4, 4, 2, 2, 1, 1, 0, 0;
14:  4, 27, 34, 22, 17, 10, 7, 4, 4, 2, 2, 1, 1, 0, 0;
15:  4, 35, 39, 33, 20, 15, 9, 7, 4, 4, 2, 2, 1, 1, 0, 0;
...
The 7 partitions of 5 and their maximal ascents are:
1:  [ 1 1 1 1 1 ]   0
2:  [ 1 1 1 2 ]   1
3:  [ 1 1 3 ]   2
4:  [ 1 2 2 ]   1
5:  [ 1 4 ]   3
6:  [ 2 3 ]   1
7:  [ 5 ]   0
There are 2 rows with 0 ascents, 3 with 1 ascent, 1 for ascents 2 and 3, giving row 5 of the triangle.
		

Links

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

Crossrefs

Cf. A238354 (partitions by minimal ascent).

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, t)+`if`(i>n, 0, (p->
      `if`(t=0 or t-i=0, p, add(coeff(p, x, j)*x^
      max(j, t-i), j=0..degree(p))))(b(n-i, i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, k), 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, If[i<1, 0, b[n, i-1, t] + If[i>n, 0, Function[{p}, If[t == 0 || t-i == 0, p, Sum[Coefficient[p, x, j]*x^ Max[j, t-i], {j, 0, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, k], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *)

Formula

G.f. for column k>=1: sum(j>=1, q^j/(1-q^j) * (prod(i=1..j-1, (1-q^((k+1)*i))/(1-q^i) ) - prod(i=1..j-1, (1-q^(k*i))/(1-q^i) ) ) ), see the comment about the g.f. in A238863.