0,4

A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.

A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 22 2007

SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

David W. Wilson, Table of n, a(n) for n = 0..300

a(5) = 4 because there are 4 complete partitions of 5: {1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 2, 2}, and {1, 1, 3}.

isCompl := proc(p, n) local m, pers, reps, f, lst, s; reps := {}; pers := combinat[permute](p); for m from 1 to nops(pers) do lst := op(m, pers); for f from 1 to nops(lst) do s := add( op(i, lst), i=1..f); reps := reps union {s}; od; od; for m from 1 to n do if not m in reps then RETURN(false); fi; od; RETURN(true); end: A126796 := proc(n) local prts, res, p; prts := combinat[partition](n); res :=0; for p from 1 to nops(prts) do if isCompl(op(p, prts), n) then res := res+1; fi; od; RETURN(res); end: for n from 1 to 40 do printf("%d %d ", n, A126796(n)); od; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 27 2007

At the beginning of the 2nd Maple program replace the current 15 by any other positive integer n in order to obtain a(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007

with(combinat): a:=proc(n) local P, b, k, p, S, j: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i], i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: end: seq(a(n), n=1..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007

with(combinat): n:=15: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i], i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: b; - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007

(PARI) T(n, k)=if(k<=1, 1, if(n<2*k-1, T(n, floor((n+1)/2)), T(n, k-1)+T(n-k, k))) a(n)=T(n, floor((n+1)/2)) /* If modified to save earlier results, this would be efficient. */ - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Mar 22 2007

Cf. A002033.

Sequence in context: A104504 A027337 A027193 this_sequence A109434 A089299 A017910

Adjacent sequences: A126793 A126794 A126795 this_sequence A126797 A126798 A126799

nonn

Brian Hopkins (bhopkins(AT)spc.edu), Feb 20 2007

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 27 2007

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2007

Further terms from Franklin T. Adams-Watters (FrankTAW@Netscape.net) and David W. Wilson, Mar 22 2007