A004250 Number of partitions of n into 3 or more parts.
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
Offset: 1
Examples
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].
From _Gus Wiseman_, Jan 18 2021: (Start)
The a(3) = 1 through a(7) = 11 graphical partitions of 2n into n parts:
(222) (2222) (22222) (222222) (2222222)
(3221) (32221) (322221) (3222221)
(33211) (332211) (3322211)
(42211) (333111) (3332111)
(422211) (4222211)
(432111) (4322111)
(522111) (4331111)
(4421111)
(5222111)
(5321111)
(6221111)
(End)
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
- 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
- Wikipedia, Starlike tree
- Index entries for sequences related to graphical partitions
Crossrefs
Programs
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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
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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 *) -
PARI
a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */
Formula
G.f.: Sum_{n>=0} (q^n / Product_{k=1..n+3} (1 - q^k)). - N. J. A. Sloane
a(n) = A000041(n) - floor((n+2)/2) = A000041(n)-A004526(n+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
a(n) = A259873(n,n). - Gus Wiseman, Jan 08 2021
Extensions
Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007
Comments