A106254 Partition table in square format.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 4, 5, 5, 3, 2, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1

Offset

1,5

Comments

The square is the following table:

1 1 1 1 1 1 1...

1 2 2 2 2 2 2...

1 2 3 3 3 3 3...

1 3 4 5 5 5 5...

1 3 5 6 7 7 7...

1 4 7 9 10 11 11...

1 4 8 11 13 14 15...

References

Ivan Niven, "Mathematics of Choice, How to Count Without Counting", MAA, 1965, pp. 98-99 (table p. 98).

Links

Martin Y. Champel, Table of n, a(n) for n = 1..405

A. Comtet, S. N. Majumdar and S. Ouvry, Integer partitions and exclusion statistics, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 11255-11269.

A. Comtet et al., Integer Partitions and Exclusion Statistics, arXiv:0705.2640 [cond-mat.stat-mech], 2007, eq. (12).

Formula

Antidiagonals of table of values of p_k(n) (the number of partitions of n with no summand greater than k).

T(n,m) = sum_{i=1..m} A008284(n,i). T(n,m) = A026820(n,m) if m<=n and T(n,m)=T(n,n) if m>=n. G.f. column m: 1/(1-x)/(1-x^2)/.../(1-x^m) = sum_(n=1,2,3..) T(n,m) x^n [Comtet]. - R. J. Mathar, Aug 31 2007

Example

The partitions of 6 are:
1 + 1 + 1 + 1 + 1 + 1; 2 + 1 + 1 + 1 + 1; 3 + 1 + 1 + 1; 2 + 2 + 1 + 1; 4 + 1 + 1; 3 + 2 + 1; 2 + 2 + 2; 5 + 1, 4 + 2, 3 + 3, 6.
There are 9 partitions of 6 having summands no larger than 4, so p_4(6) = 9.

Mathematica

T[n_, k_] := T[n, k] = If[n == 0 || k == 1, 1, T[n, k-1] + If[k > n, 0, T[n-k, k]]];
Table[T[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, after Alois P. Heinz *)

Crossrefs

Essentially the same as A008284, except for missing one diagonal thereof (which would be zero row of this array).

Sequence in context: A285521 A187451 A134542 * A117147 A111007 A176353

Adjacent sequences: A106251 A106252 A106253 * A106255 A106256 A106257

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 27 2005

Extensions

Edited, corrected and extended by Franklin T. Adams-Watters, Jan 12 2006

Status

approved