|
%I #26 Jan 12 2015 04:12:55
%S 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,
%T 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,
%U 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
%N 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.
%C Column k=0: T(n,0) = 1 + A047967(n).
%C Column k=1 is A238708.
%C Row sums are A000041.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238354/b238354.txt">Rows n = 0..140, flattened</a>
%e Triangle starts:
%e 00: 1;
%e 01: 1, 0;
%e 02: 2, 0, 0;
%e 03: 2, 1, 0, 0;
%e 04: 4, 0, 1, 0, 0;
%e 05: 5, 1, 0, 1, 0, 0;
%e 06: 8, 1, 1, 0, 1, 0, 0;
%e 07: 11, 2, 0, 1, 0, 1, 0, 0;
%e 08: 17, 2, 1, 0, 1, 0, 1, 0, 0;
%e 09: 23, 3, 1, 1, 0, 1, 0, 1, 0, 0;
%e 10: 33, 4, 2, 0, 1, 0, 1, 0, 1, 0, 0;
%e 11: 45, 5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0;
%e 12: 63, 6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0;
%e 13: 84, 8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
%e 14: 114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
%e 15: 150, 13, 4, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
%e ...
%e The 11 partitions of 6 together with their minimal ascents are:
%e 01: [ 1 1 1 1 1 1 ] 0
%e 02: [ 1 1 1 1 2 ] 0
%e 03: [ 1 1 1 3 ] 0
%e 04: [ 1 1 2 2 ] 0
%e 05: [ 1 1 4 ] 0
%e 06: [ 1 2 3 ] 1
%e 07: [ 1 5 ] 4
%e 08: [ 2 2 2 ] 0
%e 09: [ 2 4 ] 2
%e 10: [ 3 3 ] 0
%e 11: [ 6 ] 0
%e There are 8 partitions of 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.
%p b:= proc(n, i, t) option remember; `if`(n=0, 1/x, `if`(i<1, 0,
%p b(n, i-1, t)+`if`(i>n, 0, (p->`if`(t=0, p, add(coeff(
%p p, x, j)*x^`if`(j<0, t-i, min(j, t-i)),
%p j=-1..degree(p))))(b(n-i, i, i)))))
%p end:
%p T:= n->(p->seq(coeff(p, x, k)+`if`(k=0, 1, 0), k=0..n))(b(n$2, 0)):
%p seq(T(n), n=0..15);
%t 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 *)
%Y Cf. A238353 (partitions by maximal ascent).
%K nonn,tabl
%O 0,4
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 26 2014
|
