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A026799 Number of partitions of n in which the least part is 6. 18
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,19

COMMENTS

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.

LINKS

Table of n, a(n) for n=0..69.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

FORMULA

G.f.: x^6 * Product 1/(1-x^m); m=6..inf.

a(n) = p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)-p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) for n>0 where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010

EXAMPLE

a(0)=0 because there does not exist a least part of the empty partition.

The  a(6)=1 partition is 6.

The a(12)=1 partition is 6+6.

The a(13)=1 partition is 6+7.

.............................

The a(17)=1 partition is 6+11.

The a(18)=2 partitions are 6+6+6 and 6+12.

MAPLE

ZL := [ B, {B=Set(Set(Z, card>=6))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..63); # Zerinvary Lajos, Mar 13 2007

MATHEMATICA

f[1, 1] = f[0, k_] = 1; 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 6], {n, 0, 50}] (* Robert G. Wilson v, Jan 31 2011 *)

PROG

(MAGMA) p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;

A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011

CROSSREFS

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011

Sequence in context: A026824 A025149 * A185326 A238209 A210716 A027190

Adjacent sequences:  A026796 A026797 A026798 * A026800 A026801 A026802

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

STATUS

approved

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Last modified March 28 05:07 EDT 2017. Contains 284182 sequences.