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
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
Programs
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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
Comments