A290517 - OEIS

%I #24 Aug 26 2019 18:33:04

%S 1,1,1,3,4,10,60,105,280,1260,12600,27720,83160,360360,2522520,

%T 37837800,100900800,343062720,1543782240,9777287520,97772875200,

%U 2053230379200,6453009763200,24736537425600,118735379642880,742096122768000,6431499730656000

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

%H Alois P. Heinz, <a href="/A290517/b290517.txt">Table of n, a(n) for n = 0..697</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>

%F a(n) = A000142(n) / A290518(n).

%F a(0) = 1, a(n) = n * a(n-1) / A004736(n) for n>0.

%F a(n) = A309992(n,A000009(n)). - _Alois P. Heinz_, Aug 26 2019

%e 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 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,

%p      `if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))

%p     end:

%p a:= n-> n!/b(n$2):

%p seq(a(n), n=0..30);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*n/

%p       (t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 05 2017

%t 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 Cf. A000009, A000142, A000178, A004736, A022915, A290518, A309992.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Aug 04 2017