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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..697

Wikipedia, Multinomial coefficients

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 *)

Crossrefs

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

Sequence in context: A218293 A288110 A085386 * A295824 A180696 A133760

Adjacent sequences: A290514 A290515 A290516 * A290518 A290519 A290520

Keyword

nonn

Author

Alois P. Heinz, Aug 04 2017

Status

approved