A002846 Number of ways of transforming a set of n indistinguishable objects into n singletons via a sequence of n-1 refinements. (Formerly M1251 N0478)

1, 1, 1, 2, 4, 11, 33, 116, 435, 1832, 8167, 39700, 201785, 1099449, 6237505, 37406458, 232176847, 1513796040, 10162373172, 71158660160, 511957012509, 3819416719742, 29195604706757, 230713267586731, 1861978821637735, 15484368121967620, 131388840051760458

Offset

1,4

Comments

Construct the ranked poset L(n) whose nodes are the A000041(n) partitions of n, with all the partitions into the same number of parts having the same rank. A partition into k parts is joined to a partition into k+1 parts if the latter is a refinement of the former.

The partition n^1 is at the left and the partition 1^n at the right. The illustration by Olivier Gérard shows the posets L(2) through L(8).

Then a(n) is the number of paths of length n-1 in L(n) that join n^1 to 1^n.

Stated another way, a(n) is the number of maximal chains in the ranked poset L(n). (This poset is not a lattice for n > 4.) - Comments corrected by Gus Wiseman, May 01 2016

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..80

P. Erdős, R. K. Guy and J. W. Moon, On refining partitions, J. London Math. Soc., 9 (1975), 565-570.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]

Olivier Gérard, The ranked posets L(2),...,L(8)

Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9

Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 1, [Cached copy, with permission]

Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 2, [Cached copy, with permission]

Example

a(5) = 4 because there are 4 paths from top to bottom in this lattice:
  .
       ooooo
     /      \
  o.oooo   oo.ooo
    |    X    |
  o.o.ooo  o.oo.oo
     \       /
      o.o.o.oo
          |
      o.o.o.o.o
  .
(This is the ranked poset L(5), but drawn vertically rather than horizontally.)

Maple

v:= l-> [seq(`if`(i=1 or l[i]>l[i-1], seq(subs(1=[][], sort(subsop(
         i=[j, l[i]-j][], l))), j=1..l[i]/2), [][]), i=1..nops(l))]:
b:= proc(l) option remember; `if`(max(l)<2, 1, add(b(h), h=v(l))) end:
a:= n-> b([n]):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 22 2019

Mathematica

<<posets.m Table[Build[NumP[n], np]; Last@MaximalChainsDown@np, {n, 1, 25}] (* Mitch Harris, Jan 19 2006 *)

Prog

(Sage) def A002846(n): return Posets.IntegerPartitions(n).chain_polynomial().leading_coefficient()  # Max Alekseyev, Dec 23 2015

Crossrefs

See A213242, A213385, A213427 for related sequences, A327643.

Sequence in context: A268326 A268320 A127782 * A188478 A302547 A123444

Adjacent sequences: A002843 A002844 A002845 * A002847 A002848 A002849

Keyword

nonn,nice

Author

N. J. A. Sloane. Entry revised by N. J. A. Sloane, Jun 11 2012

Extensions

a(17)-a(25) from Mitch Harris, Jan 19 2006

Status

approved