A116598 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 7, 4, 4, 2, 2, 1, 1, 0, 1, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 24, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1

Offset: 0

Author

Emeric Deutsch, Feb 18 2006

Keywords

• nonn
• tabl

Comments

Row sums yield the partition numbers (A000041).

Reversed rows converge to A002865. - Joerg Arndt, Jul 07 2014

T(n,k) is the number of partitions of n for which the difference between the two largest, not necessarily distinct, parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(6,2) = 2 because we have [3,1,1,1] and [4,2]. - Emeric Deutsch, Dec 05 2015

Examples

			T(6,2) = 2 because we have [4,1,1] and [2,2,1,1].
Triangle starts:
00:   1,
01:   0,  1,
02:   1,  0,  1,
03:   1,  1,  0,  1,
04:   2,  1,  1,  0,  1,
05:   2,  2,  1,  1,  0,  1,
06:   4,  2,  2,  1,  1,  0,  1,
07:   4,  4,  2,  2,  1,  1,  0,  1,
08:   7,  4,  4,  2,  2,  1,  1,  0,  1,
09:   8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
10:  12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
11:  14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
12:  21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
13:  24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
14:  34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
15:  41, 34, 24, 21, 14, 12,  8,  7,  4,  4,  2,  2,  1,  1,  0,  1,
...
		

Links

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

Crossrefs

Cf. A000041, A002865, A000070.

Programs

Maple

with(combinat): T:=proc(n,k) if k
				

Mathematica

nn = 20; p = Product[1/(1 - x^i), {i, 2, nn}]; Prepend[CoefficientList[Table[Coefficient[Series[p /(1 - x y), {x, 0, nn}], x^n], {n, 1, nn}], y], 1] // Flatten  (* Geoffrey Critzer, Jan 22 2012 *)

Formula

G.f.: G(t,x) = 1/( (1-t*x)*prod(j>=2, 1-x^j ) ).

T(n,k) = p(n-k)-p(n-k-1) for kA000041).

T(n,0) = A002865(n).

Sum(k*T(n,k),k=0..n) = A000070(n-1) for n>=1.

Column k has g.f. x^k/prod(j>=2, 1-x^j ) (k>=0).