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 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) 43
 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 (list; graph; refs; listen; history; text; internal format)
 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 <

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Last modified December 9 18:21 EST 2022. Contains 358703 sequences. (Running on oeis4.)