A030699 Maximal value of Q(n,m) (number of partitions of n into m distinct summands) for given n.

1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1360, 1540, 1729, 1945, 2172, 2432, 2702, 3009

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• A. Comtet, S. N. Majumdar and S. Ouvry, Integer Partitions and Exclusion Statistics, arXiv:0705.2640 [cond-mat.stat-mech], 2007

Crossrefs

Cf. A003056, A008289, A257639, A260061.

Programs

Mathematica

Max /@ Table[Length@ Select[IntegerPartitions[n, m], Sort@ DeleteDuplicates@ # == Range@ m &], {n, 32}, {m, 0, n}] (* Michael De Vlieger, Nov 06 2015 *)

PARI

Q(N) = {
  my(q = vector(N)); q[1] = [1, 0, 0, 0];
  for (n = 2, N,
    my(m = (sqrtint(8*n+1) - 1)\2);
    q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
    for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
  return(q);
};
apply(vecmax, Q(59))  \\ Gheorghe Coserea, Nov 04 2015

Formula

a(n) = max {Q(n,k), k=1..m}, where m = A003056(n) and Q(n,k) is defined by A008289. - Gheorghe Coserea, Nov 04 2015

a(n) ~ K * exp(Pi*sqrt(n/3)) / n, where K = Pi / (4*sqrt(6*Pi^2 - 72*log(2)^2)) = 0.158271121170... (see A260061). - Gheorghe Coserea, Nov 08 2015

Extensions

More terms from David Wasserman, Jan 23 2002