A117468 Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part to the largest part k occurs (1<=k<=n).

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 2, 0, 0, 1, 1, 3, 2, 1, 0, 0, 1, 1, 4, 3, 1, 0, 0, 0, 1, 1, 4, 5, 1, 1, 0, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 0, 0, 1, 1, 5, 7, 3, 0, 1, 0, 0, 0, 0, 1, 1, 6, 9, 4, 1, 1, 0, 0, 0, 0, 0, 1, 1, 6, 10, 6, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 7, 12, 7, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Emeric Deutsch, Mar 19 2006

Keywords

• nonn
• tabl

Comments

Also number of partitions of n having k parts and such that all parts smaller than the largest part occur only once. Row sums yield A034296. T(n,1)=T(n,n)=1. Sum_{k=1..n} k*T(n,k) = A117469(n).

Examples

			T(10,3) = 5 because we have [3,3,2,2],[3,3,2,1,1],[3,2,2,2,1],[3,2,2,1,1,1] and [3,2,1,1,1,1,1].
Triangle starts:
1;
1,1;
1,1,1;
1,2,0,1;
1,2,1,0,1;
1,3,2,0,0,1;
		

Crossrefs

Cf. A034296, A117469.

Programs

Maple

g:=sum(t*x^j*product(1+t*x^i,i=1..j-1)/(1-t*x^j),j=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n, k) option remember; `if`(k<1 or k>n, 0,
      `if`(n=k, 1, T(n-k, k) +T(n-k, k-1)))
    end:
seq(seq(T(n,k), k=1..n), n=1..20);  # Alois P. Heinz, Jul 19 2016

Mathematica

Table[Count[IntegerPartitions@ n, m_ /; And[SubsetQ[m, Range[Min@ m, k]], Max@ m <= k]], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)
T[, 1] = 1; T[n, n_] = 1; T[n_, k_] /; 1, ] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 29 2016, adapted from Maple *)

Formula

G.f.: G(t,x) = Sum_{j>0} t*x^j*(Product_{i=1..j-1} 1+t*x^i)/(1-t*x^j).

T(n,k) = T(n-k,k) + T(n-k,k-1) where T(n,n)=1 and T(n,k)=0 when nTricia Muldoon Brown, Jul 19 2016