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A004250
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Number of partitions of n into 3 or more parts.
(Formerly M1046)
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16
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0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, 127, 168, 222, 288, 375, 480, 616, 781, 990, 1243, 1562, 1945, 2422, 2996, 3703, 4550, 5588, 6826, 8332, 10126, 12292, 14865, 17958, 21618, 25995, 31165, 37317, 44562
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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REFERENCES
| N. Metropolis and P. R. Stein, The enumeration of graphical partitions, Europ. J. Combin., 1 (1980), 139-1532.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).
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LINKS
| T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995)
Index entries for sequences related to graphical partitions
Eric Weisstein's World of Mathematics. Spider
Eric Weisstein's World of Mathematics, Spider
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FORMULA
| G.f.: sum(q^n / product( 1-q^k, k=1..n+3), n=0..inf) [ N. J. A. Sloane (njas(AT)research.att.com) ].
a(n) = A000041(n)-floor((n+2)/2) = A058984(n)-1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 18 2003
Let P(n,i) denote the number of partitions of n into i parts. Then a(n) = sum_{i=3..n} P(n,i). - Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 01 2007
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EXAMPLE
| a(6)=7 because there are three partitions of n=6 with i=3 parts: [4, 1, 1], [3, 2, 1], [2, 2, 2] and two partitions with i=4 parts: [3, 1, 1, 1], [2, 2, 1, 1] and one partition with i=5 parts: [2, 1, 1, 1, 1]] and one partition with i=6 parts: [1, 1, 1, 1, 1, 1].
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MAPLE
| Maple program from Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 01 2007: (Start)
with(combinat);
for i from 1 to 15 do pik(i, 3) od;
pik:= proc(n::integer, k::integer)
# thomas.wieder(AT)t-online.de, 30.01.07
local i, Liste, Result;
if k > n or n < 0 or k < 1 then
return fail
end if;
Result := 0;
for i from k to n do
Liste:= PartitionList(n, i);
#print(Liste);
Result := Result + nops(Liste);
end do;
return Result;
end proc;
PartitionList := proc (n, k)
# Authors: Herbert S. Wilf and Joanna Nordlicht. Source: Lecture Notes "East Side West Side, ..." University of Pennsylvania, USA, 2002. Avalible at: http://www.cis.upenn.edu/~wilf/lecnotes.html Calculates the partition of n into k parts. E.g. PartitionList(5, 2) --> [[4, 1], [3, 2]].
local East, West;
if n < 1 or k < 1 or n < k then
RETURN([])
elif n = 1 then
RETURN([[1]])
else if n < 2 or k < 2 or n < k then
West := []
else
West := map(proc (x) options operator, arrow;
[op(x), 1] end proc, PartitionList(n-1, k-1)) end if;
if k <= n-k then
East := map(proc (y) options operator, arrow;
map(proc (x) options operator, arrow; x+1 end proc, y) end proc, PartitionList(n-k, k))
else East := [] end if;
RETURN([op(West), op(East)])
end if;
end proc; # (End)
ZL :=[S, {S = Set(Cycle(Z), 3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2008
B:=[S, {S = Set(Sequence(Z, 1 <= card), card >=3)}, unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2009]
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MATHEMATICA
| Length /(AT) Table[Select[Partitions[n], Length[ # ] >= 3 &], {n, 20}] - Eric Weisstein (eric(AT)weisstein.com), May 16 2007
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CROSSREFS
| Cf. A000569, A004251, A029889, A035300, A095268.
Sequence in context: A007000 A073472 A096914 * A194805 A084842 A096967
Adjacent sequences: A004247 A004248 A004249 * A004251 A004252 A004253
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Definition corrected by Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 01 2007 and by Eric Weisstein (eric(AT)weisstein.com), May 16 2007
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