1,4

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).

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

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

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].

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]

Length /(AT) Table[Select[Partitions[n], Length[ # ] >= 3 &], {n, 20}] - Eric Weisstein (eric(AT)weisstein.com), May 16 2007

Cf. A000569, A004251, A029889, A035300, A095268.

Sequence in context: A007000 A073472 A096914 this_sequence A084842 A096967 A117276

Adjacent sequences: A004247 A004248 A004249 this_sequence A004251 A004252 A004253

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

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