login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004250 Number of partitions of n into 3 or more parts.
(Formerly M1046)
21
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; text; internal format)
OFFSET

1,4

COMMENTS

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014

REFERENCES

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).

LINKS

Table of n, a(n) for n=1..41.

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995)

N. Metropolis and P. R. Stein, The enumeration of graphical partitions, Europ. J. Combin., 1 (1980), 139-1532.

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). [Annotated scanned copy]

Eric Weisstein's World of Mathematics. Spider Graph

Index entries for sequences related to graphical partitions

FORMULA

G.f.: sum(q^n / product( 1-q^k, k=1..n+3), n=0..inf) [ N. J. A. Sloane ].

a(n) = A000041(n)-floor((n+2)/2) = A058984(n)-1. - Vladeta Jovovic, 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, Feb 01 2007

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].

MAPLE

with(combinat);

for i from 1 to 15 do pik(i, 3) od;

pik:= proc(n::integer, k::integer)

# Thomas Wieder, Jan 30 2007

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.

# Available 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;

#  Thomas Wieder, Feb 01 2007

ZL :=[S, {S = Set(Cycle(Z), 3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); # Zerinvary Lajos, Mar 25 2008

B:=[S, {S = Set(Sequence(Z, 1 <= card), card >=3)}, unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # Zerinvary Lajos, Mar 21 2009

MATHEMATICA

Length /@ Table[Select[Partitions[n], Length[#] > 2 &], {n, 20}] (* Eric W. Weisstein, May 16 2007 *)

Table[Count[Length /@ Partitions[n], _?(# > 2 &)], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

Table[PartitionsP[n] - Floor[n/2] - 1, {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

Length /@ Table[IntegerPartitions[n, {3, n}], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

PROG

(PARI) a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */

CROSSREFS

Cf. A000569, A004251, A029889, A035300, A095268, A058984.

Sequence in context: A007000 A073472 A096914 * A289060 A194805 A084842

Adjacent sequences:  A004247 A004248 A004249 * A004251 A004252 A004253

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 14:25 EDT 2018. Contains 316281 sequences. (Running on oeis4.)