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

1,5

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A003056, A008289, A257639, A260061.

Sequence in context: A367694 A358151 A152162 * A363959 A366387 A083802

Adjacent sequences: A030696 A030697 A030698 * A030700 A030701 A030702

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David Wasserman, Jan 23 2002

Status

approved