A106254 - OEIS

%I #22 Aug 08 2018 04:30:04

%S 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,

%T 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,

%U 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

%N Partition table in square format.

%C The square is the following table:

%C 1 1 1 1 1 1 1...

%C 1 2 2 2 2 2 2...

%C 1 2 3 3 3 3 3...

%C 1 3 4 5 5 5 5...

%C 1 3 5 6 7 7 7...

%C 1 4 7 9 10 11 11...

%C 1 4 8 11 13 14 15...

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

%H Martin Y. Champel, <a href="/A106254/b106254.txt">Table of n, a(n) for n = 1..405</a>

%H A. Comtet, S. N. Majumdar and S. Ouvry, <a href="http://dx.doi.org/10.1088/1751-8113/40/37/004">Integer partitions and exclusion statistics</a>, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 11255-11269.

%H A. Comtet et al., <a href="https://arxiv.org/abs/0705.2640">Integer Partitions and Exclusion Statistics</a>, arXiv:0705.2640 [cond-mat.stat-mech], 2007, eq. (12).

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

%F 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

%e The partitions of 6 are:

%e 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.

%e There are 9 partitions of 6 having summands no larger than 4, so p_4(6) = 9.

%t T[n_, k_] := T[n, k] = If[n == 0 || k == 1, 1, T[n, k-1] + If[k > n, 0, T[n-k, k]]];

%t Table[T[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 08 2018, after _Alois P. Heinz_ *)

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

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, Apr 27 2005

%E Edited, corrected and extended by _Franklin T. Adams-Watters_, Jan 12 2006