login
A290517
Maximum value of the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts.
3
1, 1, 1, 3, 4, 10, 60, 105, 280, 1260, 12600, 27720, 83160, 360360, 2522520, 37837800, 100900800, 343062720, 1543782240, 9777287520, 97772875200, 2053230379200, 6453009763200, 24736537425600, 118735379642880, 742096122768000, 6431499730656000
OFFSET
0,4
FORMULA
a(n) = A000142(n) / A290518(n).
a(0) = 1, a(n) = n * a(n-1) / A004736(n) for n>0.
a(n) = A309992(n,A000009(n)). - Alois P. Heinz, Aug 26 2019
EXAMPLE
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.
MAPLE
b:= proc(n, i) option remember; `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!)))
end:
a:= n-> n!/b(n$2):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*n/
(t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2017
MATHEMATICA
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 *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 04 2017
STATUS
approved