A002569 Max_{k=0..n} { Number of partitions of n into exactly k parts }.

1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 18, 23, 30, 37, 47, 58, 71, 90, 110, 136, 164, 201, 248, 300, 364, 436, 525, 638, 764, 919, 1090, 1297, 1549, 1845, 2194, 2592, 3060, 3590, 4242, 5013, 5888, 6912, 8070, 9418, 11004, 12866, 15021, 17475, 20298, 23501, 27169

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert Israel, Table of n, a(n) for n = 0..3000 (n = 0..97 from Robert G. Wilson v)
• F. C. Auluck, S. Chowla and H. Gupta, On the maximum value of the number of partitions into k parts, J. Indian Math. Soc., 6 (1942), 105-112. [Annotated scanned copy. But the last page is in a separate file: see the next link.]
• F. C. Auluck, S. Chowla and H. Gupta, On the maximum value of the number of partitions into k parts, [Annotated scanned copy of page 112 only]
• Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
• T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176 (p. 172, gives a(9) incorrectly as 6). [Annotated, scanned copy]
• OEIS Wiki, Sorting numbers

Crossrefs

Cf. A000041, A008284, A026819, A046155.

Programs

Maple

1, seq(max(seq(combinat:-numbpart(n,k)-combinat:-numbpart(n,k-1),k=1..n)),n=1..100); # Robert Israel, Nov 24 2014

Mathematica

f[n_] := Block[{k = 1, mx = 0}, While[k < n + 1, a = Length@ IntegerPartitions[n, {k}]; If[a > mx, mx = a]; k++ ]; mx]; Array[f, 53] (* Robert G. Wilson v, Jul 20 2010 *)
t[0, k_] := 1; t[1, k_] := 1 /; k > 0; t[n_, k_] := 0 /; n < 0; t[n_, 0] := 0 /; n > 0; t[n_, 1] := 1 /; n > 0; t[n_, k_] := t[n, k] = Sum[t[n - k + i, k - i], {i, 0, k - 1}];
f[n_] := Max[ Table[ t[n - k, k], {k, 0, n}]]; Array[f, 54, 0] (* Robert G. Wilson v, Nov 24 2014 *)
Max[CoefficientList[#, a]] & /@ (1/QPochhammer[a q, q] + O[q]^60)[[3]] (* Vladimir Reshetnikov, Nov 17 2016 *)

Extensions

More terms from David W. Wilson