A290517 Maximum value of the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts.
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
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..697
- Wikipedia, Multinomial coefficients
Programs
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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 *)