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A290517 Maximum value of the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts.

%I A290517 #24 Aug 26 2019 18:33:04
%S A290517 1,1,1,3,4,10,60,105,280,1260,12600,27720,83160,360360,2522520,
%T A290517 37837800,100900800,343062720,1543782240,9777287520,97772875200,
%U A290517 2053230379200,6453009763200,24736537425600,118735379642880,742096122768000,6431499730656000
%N A290517 Maximum value of the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts.
%H A290517 Alois P. Heinz, <a href="/A290517/b290517.txt">Table of n, a(n) for n = 0..697</a>
%H A290517 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%F A290517 a(n) = A000142(n) / A290518(n).
%F A290517 a(0) = 1, a(n) = n * a(n-1) / A004736(n) for n>0.
%F A290517 a(n) = A309992(n,A000009(n)). - _Alois P. Heinz_, Aug 26 2019
%e A290517 a(10) = 12600 = 10! / (4! * 3! * 2! * 1!) is the value for partition [4,3,2,1]. All other partitions of 10 into distinct parts give smaller values: [5,3,2]-> 2520, [5,4,1]-> 1260, [6,3,1]-> 840, [6,4]-> 210, [7,2,1]-> 360, [7,3]-> 120, [8,2]-> 45, [9,1]-> 10, [10]-> 1.
%p A290517 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,
%p A290517      `if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))
%p A290517     end:
%p A290517 a:= n-> n!/b(n$2):
%p A290517 seq(a(n), n=0..30);
%p A290517 # second Maple program:
%p A290517 a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*n/
%p A290517       (t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
%p A290517     end:
%p A290517 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 05 2017
%t A290517 b[n_, i_]:=b[n, i]=If[n>i*(i + 1)/2, Infinity, If[n==0, 1, Min[b[n, i - 1], b[n - i, Min[n - i, i - 1]]*i!]]]; Table[n!/b[n, n], {n, 0, 30}] (* _Indranil Ghosh_, Aug 05 2017, after Maple *)
%Y A290517 Cf. A000009, A000142, A000178, A004736, A022915, A290518, A309992.
%K A290517 nonn
%O A290517 0,4
%A A290517 _Alois P. Heinz_, Aug 04 2017