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A102356 Problem 66 in Knuth's Art of Computer Programming, vol. 4, section 7.2.1.5 asks which integer partition of n produces the most set partitions. The n-th term of this sequence is the number of set partitions produced by that integer partition. 7
1, 1, 1, 3, 6, 15, 60, 210, 840, 3780, 12600, 69300, 415800, 2702700, 12612600, 94594500, 756756000, 4288284000, 38594556000, 244432188000, 1833241410000, 17110253160000, 141159588570000, 1298668214844000, 10389345718752000, 108222351237000000, 1125512452864800000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the maximum value in row n of A080575.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

D. E. Knuth, The Art of Computer Programming, vol. 4. See Section 7.2.1.5, Problem 66, pages 439 and 778.

EXAMPLE

a(4) = 6 because there are 6 set partitions of type {2,1,1}, namely 12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34; all other integer partitions of 4 produce fewer set partitions.

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

       max(seq(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i))))

    end:

a:= n-> b(n, n):

seq(a(n), n=0..40);  # Alois P. Heinz, Apr 13 2012

MATHEMATICA

sp[l_] := (Total[l])!/(Apply[Times, Map[ #! &, l]]*Apply[Times, Map[Count[l, # ]! &, Range[Max[l]]]]) a[n_] := Max[Map[sp, Partitions[n]]]

b[0, _] = 1; b[_, _?NonPositive] = 0; b[n_, i_] := b[n, i] = Max[Table[ b[n - i*j, i-1]*n!/i!^j/(n - i*j)!/j!, {j, 0, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Jan 24 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A080575, A102456.

Sequence in context: A230950 A267552 A241269 * A208662 A102936 A009192

Adjacent sequences:  A102353 A102354 A102355 * A102357 A102358 A102359

KEYWORD

nonn

AUTHOR

Dan Drake, Feb 21 2005

EXTENSIONS

More terms from Alois P. Heinz, Oct 13 2011.

Typo in definition corrected by Klaus Leeb, Apr 30 2014.

STATUS

approved

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Last modified May 24 17:09 EDT 2019. Contains 323533 sequences. (Running on oeis4.)