cp's OEIS Frontend

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

%I A030699 #49 Nov 03 2023 06:50:49
%S A030699 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,
%T A030699 84,94,108,120,136,150,169,192,221,255,291,333,377,427,480,540,603,
%U A030699 674,748,831,918,1014,1115,1226,1360,1540,1729,1945,2172,2432,2702,3009
%N A030699 Maximal value of Q(n,m) (number of partitions of n into m distinct summands) for given n.
%D A030699 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
%H A030699 Gheorghe Coserea, <a href="/A030699/b030699.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A030699 A. Comtet, S. N. Majumdar and S. Ouvry, <a href="http://arXiv.org/abs/0705.2640">Integer Partitions and Exclusion Statistics</a>, arXiv:0705.2640 [cond-mat.stat-mech], 2007
%F A030699 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
%F A030699 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
%t A030699 Max /@ Table[Length@ Select[IntegerPartitions[n, m], Sort@ DeleteDuplicates@ # == Range@ m &], {n, 32}, {m, 0, n}] (* _Michael De Vlieger_, Nov 06 2015 *)
%o A030699 (PARI)
%o A030699 Q(N) = {
%o A030699   my(q = vector(N)); q[1] = [1, 0, 0, 0];
%o A030699   for (n = 2, N,
%o A030699     my(m = (sqrtint(8*n+1) - 1)\2);
%o A030699     q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
%o A030699     for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
%o A030699   return(q);
%o A030699 };
%o A030699 apply(vecmax, Q(59))  \\ _Gheorghe Coserea_, Nov 04 2015
%Y A030699 Cf. A003056, A008289, A257639, A260061.
%K A030699 nonn,easy
%O A030699 1,5
%A A030699 _N. J. A. Sloane_
%E A030699 More terms from _David Wasserman_, Jan 23 2002