# Q. about triangle A106239 (graphs with unicyclic components)

Gabriel Cunningham gabriel.cunningham at gmail.com
Mon Feb 25 03:39:28 CET 2008

The example given says:
%e A106239 T(6,2)=10 because there are 10 such graphs of order 6 with 2
components. The value of T(n,m) is zero if and only if m > floor(n/3).

The table does give T(5,2)=0. It is T(6,2) that is 10; the offset is 1, so
the first entry is T(1, 1).

Gabe

On Sun, Feb 24, 2008 at 9:07 PM, N. J. A. Sloane <njas at research.att.com>
wrote:

>
> Dear Seqfans and Associate OEIS Editors:
>
> Here is the current entry for A106239:
>
> %I A106239
> %S A106239
> 0,0,0,1,0,0,15,0,0,0,222,0,0,0,0,3660,10,0,0,0,0,68295,525,0,0,0,0,0,
> %T A106239
> 1436568,20307,0,0,0,0,0,0,33779340,727020,280,0,0,0,0,0,0,880107840,
> %U A106239
> 25934184,31500,0,0,0,0,0,0,0,25201854045,950478210,2325015,0,0,0,0,0,0
> %N A106239 Triangle read by rows: T(n,m) = number of graphs on n labeled
> nodes, with m components of size = order. Also number of graphs on n labeled
> nodes with m unicyclic components.
> %F A106239 E.g.f.:
> Jovovic (vladeta(AT)Eunet.yu), May 04 2005
> %F A106239 T(n, m)= sum N/D over the partitions of n:1K1+2K2+ ... +nKn, =
> with exactly m parts greater than 2, where N = n!*product_{1=<i<=n}=
> A057500(i)^Ki, and D = product_{1=<i<=n}(Ki!(i!)^Ki).
> %e A106239 T(6,2)=10 because there are 10 such graphs of order 6 with 2
> components. The value of T(n,m) is zero if and only if m > floor(n/3).
> %Y A106239 Cf. A057500 and A106238 (similar formulae that can be used in
> the unlabeled case.).
> %K A106239 nonn,tabl
> %O A106239 1,7
> %A A106239 Washington Bomfim (webonfim(AT)bol.com.br), May 03 2005
>
> There are three other mentions of it:
>
> %C A137916 The first values are row sums of A106239.
> %Y A106238 Cf. A057276, A035512, A106237, A106239.
> %H A106240 Washington Bomfim, <a href="
> http://webonfim.vilabol.uol.com.br/graphsunicycliccomponents.html">Illustration
> of A106239</a>
>
> The other day Tom Zaslavsky, zaslav at math.binghamton.edu, wrote to me
> saying:
>
> %N A106239 T(n,m) = # of simple graphs on n labeled nodes with m
> components, each component being unicyclic (one cycle).
> The largest possible number of components is floor(n/3), so T(5,2) = 0,
> but the table shows 10.
> The nonzero lower boundary should have slope -1/3, but its slope is -1/2.
> Am I being completely stupid, or is there something wrong?
> I'm interested because I'm editing Wikipedia on "pseudoforests" and \sum_m
> T(n,m) (which is apparently not in the database) should be the number of
> simple pseudoforests on n labelled vertices.
>
> Can someone explain what this sequence is showing? Vladeta?
> Thanks!
> Neil
>
>
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