[seqfan] About empty graphs

Wed Aug 19 02:19:08 CEST 2015

First, I'll complain about the terminology.  Although many people call
the graph with no vertices an "empty graph", a much larger number
of people use "empty graph" to mean a graph with no edges (and
any number of vertices, i.e. the complement of a complete graph).
The ratio must be 10:1 at least (search in Google Scholar).
A much less confusing name for the graph with none of anything
is "null graph".

As to what properties graph theorists award to the null graph,
the answer is:

(1) For 99% of applications the null graph is ignored completely.
If you read a paper in which the null graph is not mentioned,
it is safest to assume that the author doesn't even believe in
null graphs. You absolutely must not assume the theorems
work on null graphs unless the paper says they do.

(2) If it is convenient to admit the null graph because it
simplifies some theory, it is given whatever properties work
best in that situation.  For example, in enumeration the
question often arises of what is the constant term in some
generating function.  A classic lovely theorem is that if
C(x) is the exponential generating function for connected
graphs, then G(x) = exp(C(x)) is the exponential generating
function for all graphs.  For this to work, C(x) must have
zero constant term, which causes G(x) to have 1 as the
constant term.  So in this case if you want to interpret
the constant terms as referring to the null graph, there
is exactly one of them and it is not connected.  Some
authors who are null-graph-atheists write exp(C(x))-1.

I'm sure there are other examples that come out nicer
if the null graph is connected, but probably more things
"work out" if the null graph is not connected. I don't see
any point in playing with the wording of definitions to
make it come out one way or another.

Is the null graph a tree?  Hint: remember the wonderful
formula n^(n-2) for the number of trees with n vertices.

Cheers, Brendan.

On 19/08/2015 7:43 am, seqfan-request at list.seqfan.eu wrote:
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> Today's Topics:
>
>     1. Re: Number of connected simplicial complexes (Charles Greathouse)
>     2. Re: Number of connected simplicial complexes (Frank Adams-Watters)
>     3. Re: Number of connected simplicial complexes (ALLOUCHE Jean-paul)
>     4. Re: Number of connected simplicial complexes (M. F. Hasler)
>     5. Re: Number of connected simplicial complexes (Benoît Jubin)
>     6. Re: Number of connected simplicial complexes (ALLOUCHE Jean-paul)
>     7. Re: Number of connected simplicial complexes (Max Alekseyev)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Tue, 18 Aug 2015 09:51:30 -0400
> From: Charles Greathouse <charles.greathouse at case.edu>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<CAAkfSGJEk1PzwtfdirB2xpVZ5cMxKGj3oi9v8c1KOUecY+nSyQ at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> I'm curious, what are the arguments for the empty graph being disconnected?
> For every pair (x, y) of distinct vertices there is an edge between them.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Benoit,
>> Very nice! I'll update the OEIS accordingly. The two
>> new entries will be A261005, A261006.
>>
>> (I don't feel strongly about it, but one could
>> argue that the empty graph IS connected,
>> as in A001349, see the comments there,
>> since "the empty set has every property".
>> I think the arguments for and against are about equally strong!)
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>> wrote:
>>
>>> Dear Neil,
>>>
>>> I would say that the next term is 157. Indeed, the number of simplicial
>>> complexes with n vertices is (starting at n=0):
>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>> It is in the OEIS... and well hidden in it. It is A006602, see comment
>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>> deserves
>>> its own entry.
>>> Then, to count the connected ones, there is the usual summation over
>>> partitions:
>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots} \prod_{i=1}^p
>>> a_{conn}(n_i)
>>> so the connected version is given by
>>> 0, 1, 1, 3, 14, 157, ...
>>> and the next two terms can be easily computed.
>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>> connected.
>>>
>>> Best regards,
>>> Benoît
>>>
>>>
>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>> sent me the initial terms of four sequences that arise in
>>>> the enumeration of simplicial complexes. The second one
>>>> is now A048143, and you can see an annotated
>>>> and corrected scan of his letter there.
>>>> But what about the first one? This is the number of
>>>> unlabeled connected simplicial complexes on n nodes. It
>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>> If someone could find one or two more terms, we could add it to the
>> OEIS.
>>>> Vladeta Jovovic, where are you when we need you?
>>>>
>>>> Greg also mentions two other sequences arising from his study
>>>> of the early universe, but their definitions are not very explicit.
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> ------------------------------
>
> Message: 2
> Date: Tue, 18 Aug 2015 11:26:21 -0400
> From: Frank Adams-Watters <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID: <14f416b17a6-ce3-11561 at webprd-a57.mail.aol.com>
> Content-Type: text/plain; charset=utf-8
>
> The way I look at it, it's not so much that it's disconnected, as that connected is not quite the right concept. Instead, one is usually interested in single-component graphs - there exists a node from which you can trace a path to every other node, rather than for all nodes you can trace a path to every other node.
>
> The reason for preferring this is similar to why 1 is not considered a prime: every graph has a unique decomposition into single-component graphs.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Charles Greathouse <charles.greathouse at case.edu>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Tue, Aug 18, 2015 8:52 am
> Subject: [seqfan] Re: Number of connected simplicial complexes
>
>
> I'm curious, what are the arguments for the empty graph being disconnected?
> For
> every pair (x, y) of distinct vertices there is an edge between them.
>
> Charles
> Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Thu, Aug 13,
> 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Benoit,
>> Very
> nice! I'll update the OEIS accordingly. The two
>> new entries will be A261005,
> A261006.
>> (I don't feel strongly about it, but one could
>> argue that the
> empty graph IS connected,
>> as in A001349, see the comments there,
>> since "the
> empty set has every property".
>> I think the arguments for and against are about
> equally strong!)
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS
> Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also
> Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone:
> 732 828 6098; home page: http://NeilSloane.com
>> Email:
> njasloane at gmail.com
>>
>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin
> <benoit.jubin at gmail.com>
>> wrote:
>>
>>> Dear Neil,
>>>
>>> I would say that the
> next term is 157. Indeed, the number of simplicial
>>> complexes with n vertices
> is (starting at n=0):
>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>> It is
> in the OEIS... and well hidden in it. It is A006602, see comment
>>> by...
> Vladeta Jovovic. Since the zeroth term differs, it probably
>> deserves
>>> its
> own entry.
>>> Then, to count the connected ones, there is the usual summation
> over
>>> partitions:
>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2
> \dots} \prod_{i=1}^p
>>> a_{conn}(n_i)
>>> so the connected version is given
> by
>>> 0, 1, 1, 3, 14, 157, ...
>>> and the next two terms can be easily
> computed.
>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is
> not
>>> connected.
>>>
>>> Best regards,
>>> Benoît
>>>
>>>
>>> On Wed, Jul 8,
> 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Dear Seq
> Fans, Back in 1983 the physicist Greg Huber
>>>> sent me the initial terms of
> four sequences that arise in
>>>> the enumeration of simplicial complexes. The
> second one
>>>> is now A048143, and you can see an annotated
>>>> and
> corrected scan of his letter there.
>>>> But what about the first one? This is
> the number of
>>>> unlabeled connected simplicial complexes on n nodes. It
>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>> If
> someone could find one or two more terms, we could add it to the
>> OEIS.
>>> Vladeta Jovovic, where are you when we need you?
>>>> Greg also
> mentions two other sequences arising from his study
>>>> of the early universe,
> but their definitions are not very explicit.
>>>>
> _______________________________________________
>>>> Seqfan Mailing list
> - http://list.seqfan.eu/
>>>
> _______________________________________________
>>> Seqfan Mailing list -
> http://list.seqfan.eu/
>>
> _______________________________________________
>> Seqfan Mailing list -
> http://list.seqfan.eu/
> _______________________________________________
>
> Seqfan
> Mailing list - http://list.seqfan.eu/
>
>
>
>
> ------------------------------
>
> Message: 3
> Date: Tue, 18 Aug 2015 19:41:26 +0000
> From: ALLOUCHE Jean-paul <Jean-paul.ALLOUCHE at imj-prg.fr>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<6DD048801492FC45ACCDF9FE005AAB2216017B88 at CNMB02WVP.core-res.rootcore.local>
> 	
> Content-Type: text/plain; charset="iso-8859-1"
>
> Well it is also true that for every pair (x,y) of distinct vertices there is no
> edge between them (there are no such pairs, so it is equally true that there
> is no vertex or that there is a vertex). As formulated "intuitively" by some
> people "the empty set has all properties".
>
> not quite easy to get convinced after all, but...
>
> best
> jean-paul
> ________________________________________
> De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Charles Greathouse [charles.greathouse at case.edu]
> Envoyé : mardi 18 août 2015 15:51
> À : Sequence Fanatics Discussion list
> Objet : [seqfan] Re: Number of connected simplicial complexes
>
> I'm curious, what are the arguments for the empty graph being disconnected?
> For every pair (x, y) of distinct vertices there is an edge between them.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Benoit,
>> Very nice! I'll update the OEIS accordingly. The two
>> new entries will be A261005, A261006.
>>
>> (I don't feel strongly about it, but one could
>> argue that the empty graph IS connected,
>> as in A001349, see the comments there,
>> since "the empty set has every property".
>> I think the arguments for and against are about equally strong!)
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>> wrote:
>>
>>> Dear Neil,
>>>
>>> I would say that the next term is 157. Indeed, the number of simplicial
>>> complexes with n vertices is (starting at n=0):
>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>> It is in the OEIS... and well hidden in it. It is A006602, see comment
>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>> deserves
>>> its own entry.
>>> Then, to count the connected ones, there is the usual summation over
>>> partitions:
>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots} \prod_{i=1}^p
>>> a_{conn}(n_i)
>>> so the connected version is given by
>>> 0, 1, 1, 3, 14, 157, ...
>>> and the next two terms can be easily computed.
>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>> connected.
>>>
>>> Best regards,
>>> Benoît
>>>
>>>
>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>> sent me the initial terms of four sequences that arise in
>>>> the enumeration of simplicial complexes. The second one
>>>> is now A048143, and you can see an annotated
>>>> and corrected scan of his letter there.
>>>> But what about the first one? This is the number of
>>>> unlabeled connected simplicial complexes on n nodes. It
>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>> If someone could find one or two more terms, we could add it to the
>> OEIS.
>>>> Vladeta Jovovic, where are you when we need you?
>>>>
>>>> Greg also mentions two other sequences arising from his study
>>>> of the early universe, but their definitions are not very explicit.
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
> ------------------------------
>
> Message: 4
> Date: Tue, 18 Aug 2015 16:11:04 -0400
> From: "M. F. Hasler" <oeis at hasler.fr>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<CAOPi3QKjqJOHP3E_UC_ARR=mVMkWEjTxO8My2oW+77nY+riU0g at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> But the definition of disconnected is not "all vertices..." but "there
> exist..."
> e.g.: from mathworld:
> "A graph is said to be disconnected if it is not connected, i.e., if there
> exist two nodes such that no path..."
>
> and for the empty graph there are no such two nodes...
>
> This is a little bit different from the situation of e.g.the empty set
> which is open and closed at the same time.
>
> So I would have answered like Charles and I would be astonished if a
> "serious" reference work would consider the empty graph as not connected...
>
> Maximilian
> Le 18 août 2015 21:41, "ALLOUCHE Jean-paul" <Jean-paul.ALLOUCHE at imj-prg.fr>
> a écrit :
>
>> Well it is also true that for every pair (x,y) of distinct vertices there
>> is no
>> edge between them (there are no such pairs, so it is equally true that
>> there
>> is no vertex or that there is a vertex). As formulated "intuitively" by
>> some
>> people "the empty set has all properties".
>>
>> not quite easy to get convinced after all, but...
>>
>> best
>> jean-paul
>> ________________________________________
>> De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Charles
>> Greathouse [charles.greathouse at case.edu]
>> Envoyé : mardi 18 août 2015 15:51
>> À : Sequence Fanatics Discussion list
>> Objet : [seqfan] Re: Number of connected simplicial complexes
>>
>> I'm curious, what are the arguments for the empty graph being disconnected?
>> For every pair (x, y) of distinct vertices there is an edge between them.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:
>>
>>> Benoit,
>>> Very nice! I'll update the OEIS accordingly. The two
>>> new entries will be A261005, A261006.
>>>
>>> (I don't feel strongly about it, but one could
>>> argue that the empty graph IS connected,
>>> as in A001349, see the comments there,
>>> since "the empty set has every property".
>>> I think the arguments for and against are about equally strong!)
>>>
>>> Best regards
>>> Neil
>>>
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>>
>>>
>>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>>> wrote:
>>>
>>>> Dear Neil,
>>>>
>>>> I would say that the next term is 157. Indeed, the number of simplicial
>>>> complexes with n vertices is (starting at n=0):
>>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>>> It is in the OEIS... and well hidden in it. It is A006602, see comment
>>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>>> deserves
>>>> its own entry.
>>>> Then, to count the connected ones, there is the usual summation over
>>>> partitions:
>>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots} \prod_{i=1}^p
>>>> a_{conn}(n_i)
>>>> so the connected version is given by
>>>> 0, 1, 1, 3, 14, 157, ...
>>>> and the next two terms can be easily computed.
>>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>>> connected.
>>>>
>>>> Best regards,
>>>> Benoît
>>>>
>>>>
>>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>>> wrote:
>>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>>> sent me the initial terms of four sequences that arise in
>>>>> the enumeration of simplicial complexes. The second one
>>>>> is now A048143, and you can see an annotated
>>>>> and corrected scan of his letter there.
>>>>> But what about the first one? This is the number of
>>>>> unlabeled connected simplicial complexes on n nodes. It
>>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>>> If someone could find one or two more terms, we could add it to the
>>> OEIS.
>>>>> Vladeta Jovovic, where are you when we need you?
>>>>>
>>>>> Greg also mentions two other sequences arising from his study
>>>>> of the early universe, but their definitions are not very explicit.
>>>>>
>>>>> _______________________________________________
>>>>>
>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> ------------------------------
>
> Message: 5
> Date: Tue, 18 Aug 2015 23:19:52 +0200
> From: Benoît Jubin <benoit.jubin at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<CAEG8PYvNTyMEHq7GfZFp5hxKr84s27W9Y7h3_GT7jSJJBx4kGg at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> The question is rather: "Is it more convenient to define the empty graph as
> connected or not?" (in the sense that more theorems are then stated without
> the need to deal with special cases).
>
> As Frank Adams-Watter wrote, this is similar to 1 being prime or not:
> unique decomposition as a product of primes / as a disjoint union of
> connected components. Also, the first comment in http://oeis.org/A001349,
> "but with a(0) omitted" acknowledges that the empty graph should not be
> counted as connected: this would simplify the statements of mathematical
> results.
>
> In this respect, I recommend the reading of
> http://math.stackexchange.com/questions/50551/is-the-empty-graph-connected
> and
> http://ncatlab.org/nlab/show/too+simple+to+be+simple
> and
> http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree
>
> Regards,
> Benoît
>
>
> On Tue, Aug 18, 2015 at 10:11 PM, M. F. Hasler <oeis at hasler.fr> wrote:
>
>> But the definition of disconnected is not "all vertices..." but "there
>> exist..."
>> e.g.: from mathworld:
>> "A graph is said to be disconnected if it is not connected, i.e., if there
>> exist two nodes such that no path..."
>>
>> and for the empty graph there are no such two nodes...
>>
>> This is a little bit different from the situation of e.g.the empty set
>> which is open and closed at the same time.
>>
>> So I would have answered like Charles and I would be astonished if a
>> "serious" reference work would consider the empty graph as not connected...
>>
>> Maximilian
>> Le 18 août 2015 21:41, "ALLOUCHE Jean-paul" <Jean-paul.ALLOUCHE at imj-prg.fr
>> a écrit :
>>
>>> Well it is also true that for every pair (x,y) of distinct vertices there
>>> is no
>>> edge between them (there are no such pairs, so it is equally true that
>>> there
>>> is no vertex or that there is a vertex). As formulated "intuitively" by
>>> some
>>> people "the empty set has all properties".
>>>
>>> not quite easy to get convinced after all, but...
>>>
>>> best
>>> jean-paul
>>> ________________________________________
>>> De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Charles
>>> Greathouse [charles.greathouse at case.edu]
>>> Envoyé : mardi 18 août 2015 15:51
>>> À : Sequence Fanatics Discussion list
>>> Objet : [seqfan] Re: Number of connected simplicial complexes
>>>
>>> I'm curious, what are the arguments for the empty graph being
>> disconnected?
>>> For every pair (x, y) of distinct vertices there is an edge between them.
>>>
>>> Charles Greathouse
>>> Analyst/Programmer
>>> Case Western Reserve University
>>>
>>> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Benoit,
>>>> Very nice! I'll update the OEIS accordingly. The two
>>>> new entries will be A261005, A261006.
>>>>
>>>> (I don't feel strongly about it, but one could
>>>> argue that the empty graph IS connected,
>>>> as in A001349, see the comments there,
>>>> since "the empty set has every property".
>>>> I think the arguments for and against are about equally strong!)
>>>>
>>>> Best regards
>>>> Neil
>>>>
>>>> Neil J. A. Sloane, President, OEIS Foundation.
>>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
>> NJ.
>>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>>> Email: njasloane at gmail.com
>>>>
>>>>
>>>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>>>> wrote:
>>>>
>>>>> Dear Neil,
>>>>>
>>>>> I would say that the next term is 157. Indeed, the number of
>> simplicial
>>>>> complexes with n vertices is (starting at n=0):
>>>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>>>> It is in the OEIS... and well hidden in it. It is A006602, see
>> comment
>>>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>>>> deserves
>>>>> its own entry.
>>>>> Then, to count the connected ones, there is the usual summation over
>>>>> partitions:
>>>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots}
>> \prod_{i=1}^p
>>>>> a_{conn}(n_i)
>>>>> so the connected version is given by
>>>>> 0, 1, 1, 3, 14, 157, ...
>>>>> and the next two terms can be easily computed.
>>>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>>>> connected.
>>>>>
>>>>> Best regards,
>>>>> Benoît
>>>>>
>>>>>
>>>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>>>> wrote:
>>>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>>>> sent me the initial terms of four sequences that arise in
>>>>>> the enumeration of simplicial complexes. The second one
>>>>>> is now A048143, and you can see an annotated
>>>>>> and corrected scan of his letter there.
>>>>>> But what about the first one? This is the number of
>>>>>> unlabeled connected simplicial complexes on n nodes. It
>>>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>>>> If someone could find one or two more terms, we could add it to the
>>>> OEIS.
>>>>>> Vladeta Jovovic, where are you when we need you?
>>>>>>
>>>>>> Greg also mentions two other sequences arising from his study
>>>>>> of the early universe, but their definitions are not very explicit.
>>>>>>
>>>>>> _______________________________________________
>>>>>>
>>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>>
>>>>> _______________________________________________
>>>>>
>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
> ------------------------------
>
> Message: 6
> Date: Tue, 18 Aug 2015 21:25:36 +0000
> From: ALLOUCHE Jean-paul <Jean-paul.ALLOUCHE at imj-prg.fr>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<6DD048801492FC45ACCDF9FE005AAB2216017C40 at CNMB02WVP.core-res.rootcore.local>
> 	
> Content-Type: text/plain; charset="iso-8859-1"
>
> Yip Maximilian is right in the sense that my argument was wrong and
> that I am not an expert logician to find a better argument :-)
> On the other hand the pragmatic approach of Benoit has its advantages:
> it could probably be turned into an unambiguous `a la Bourbaki definition.
> [The case of p prime if and only if it is divisible only by itself and by 1,
> is easily "slightly" transformed to eliminate 1 by saying "if and only if
> it has exactly two divisors".]
>
> best wishes
> jean-paul
> ________________________________________
> De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Benoît Jubin [benoit.jubin at gmail.com]
> Envoyé : mardi 18 août 2015 23:19
> À : Sequence Fanatics Discussion list
> Objet : [seqfan] Re: Number of connected simplicial complexes
>
> The question is rather: "Is it more convenient to define the empty graph as
> connected or not?" (in the sense that more theorems are then stated without
> the need to deal with special cases).
>
> As Frank Adams-Watter wrote, this is similar to 1 being prime or not:
> unique decomposition as a product of primes / as a disjoint union of
> connected components. Also, the first comment in http://oeis.org/A001349,
> "but with a(0) omitted" acknowledges that the empty graph should not be
> counted as connected: this would simplify the statements of mathematical
> results.
>
> In this respect, I recommend the reading of
> http://math.stackexchange.com/questions/50551/is-the-empty-graph-connected
> and
> http://ncatlab.org/nlab/show/too+simple+to+be+simple
> and
> http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree
>
> Regards,
> Benoît
>
>
> On Tue, Aug 18, 2015 at 10:11 PM, M. F. Hasler <oeis at hasler.fr> wrote:
>
>> But the definition of disconnected is not "all vertices..." but "there
>> exist..."
>> e.g.: from mathworld:
>> "A graph is said to be disconnected if it is not connected, i.e., if there
>> exist two nodes such that no path..."
>>
>> and for the empty graph there are no such two nodes...
>>
>> This is a little bit different from the situation of e.g.the empty set
>> which is open and closed at the same time.
>>
>> So I would have answered like Charles and I would be astonished if a
>> "serious" reference work would consider the empty graph as not connected...
>>
>> Maximilian
>> Le 18 août 2015 21:41, "ALLOUCHE Jean-paul" <Jean-paul.ALLOUCHE at imj-prg.fr
>> a écrit :
>>
>>> Well it is also true that for every pair (x,y) of distinct vertices there
>>> is no
>>> edge between them (there are no such pairs, so it is equally true that
>>> there
>>> is no vertex or that there is a vertex). As formulated "intuitively" by
>>> some
>>> people "the empty set has all properties".
>>>
>>> not quite easy to get convinced after all, but...
>>>
>>> best
>>> jean-paul
>>> ________________________________________
>>> De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Charles
>>> Greathouse [charles.greathouse at case.edu]
>>> Envoyé : mardi 18 août 2015 15:51
>>> À : Sequence Fanatics Discussion list
>>> Objet : [seqfan] Re: Number of connected simplicial complexes
>>>
>>> I'm curious, what are the arguments for the empty graph being
>> disconnected?
>>> For every pair (x, y) of distinct vertices there is an edge between them.
>>>
>>> Charles Greathouse
>>> Analyst/Programmer
>>> Case Western Reserve University
>>>
>>> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Benoit,
>>>> Very nice! I'll update the OEIS accordingly. The two
>>>> new entries will be A261005, A261006.
>>>>
>>>> (I don't feel strongly about it, but one could
>>>> argue that the empty graph IS connected,
>>>> as in A001349, see the comments there,
>>>> since "the empty set has every property".
>>>> I think the arguments for and against are about equally strong!)
>>>>
>>>> Best regards
>>>> Neil
>>>>
>>>> Neil J. A. Sloane, President, OEIS Foundation.
>>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
>> NJ.
>>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>>> Email: njasloane at gmail.com
>>>>
>>>>
>>>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>>>> wrote:
>>>>
>>>>> Dear Neil,
>>>>>
>>>>> I would say that the next term is 157. Indeed, the number of
>> simplicial
>>>>> complexes with n vertices is (starting at n=0):
>>>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>>>> It is in the OEIS... and well hidden in it. It is A006602, see
>> comment
>>>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>>>> deserves
>>>>> its own entry.
>>>>> Then, to count the connected ones, there is the usual summation over
>>>>> partitions:
>>>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots}
>> \prod_{i=1}^p
>>>>> a_{conn}(n_i)
>>>>> so the connected version is given by
>>>>> 0, 1, 1, 3, 14, 157, ...
>>>>> and the next two terms can be easily computed.
>>>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>>>> connected.
>>>>>
>>>>> Best regards,
>>>>> Benoît
>>>>>
>>>>>
>>>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>>>> wrote:
>>>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>>>> sent me the initial terms of four sequences that arise in
>>>>>> the enumeration of simplicial complexes. The second one
>>>>>> is now A048143, and you can see an annotated
>>>>>> and corrected scan of his letter there.
>>>>>> But what about the first one? This is the number of
>>>>>> unlabeled connected simplicial complexes on n nodes. It
>>>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>>>> If someone could find one or two more terms, we could add it to the
>>>> OEIS.
>>>>>> Vladeta Jovovic, where are you when we need you?
>>>>>>
>>>>>> Greg also mentions two other sequences arising from his study
>>>>>> of the early universe, but their definitions are not very explicit.
>>>>>>
>>>>>> _______________________________________________
>>>>>>
>>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>>
>>>>> _______________________________________________
>>>>>
>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
> ------------------------------
>
> Message: 7
> Date: Tue, 18 Aug 2015 17:42:38 -0400
> From: Max Alekseyev <maxale at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Number of connected simplicial complexes
> Message-ID:
> 	<CAJkPp5NmbBCkAi-f16x0eiWZeMcM2EiOy3C4ryApqLmbi0n5Pg at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> I'd like to comment on "the empty set has every property". This is not
> quite true and, in fact, depends on the property. To figure out whether or
> not the empty object (in particular, the empty set) has a particular
> property, the property first needs to be formalized as a logic formula with
> quantifiers. For the empty object, every universal quantifier (∀) is
> evaluated to 'true' and every existential quantifier (∃) is evaluated to
> 'false'.
>
> For example, we may ask if the empty permutation represents a derangement.
> For permutation p being a derangement can be formalized as ∀ i∈D(p): p(i) ≠
> i, where D(p) is the permutation domain of p. For the empty permutation e,
> we have D(e) = empty set and thus "∀ i∈D(e): ..." is evaluated to 'true',
> not matter what is hidden behind "...". So the empty permutation is
> derangement.
> It can be also shown that the empty permutation represents the identity (∀
> i∈D(p): p(i)=i). We need to be careful here as no non-empty permutation can
> be derangement and identity at the same time.
>
> Regards,
> Max
>
> On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Benoit,
>> Very nice! I'll update the OEIS accordingly. The two
>> new entries will be A261005, A261006.
>>
>> (I don't feel strongly about it, but one could
>> argue that the empty graph IS connected,
>> as in A001349, see the comments there,
>> since "the empty set has every property".
>> I think the arguments for and against are about equally strong!)
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
>> wrote:
>>
>>> Dear Neil,
>>>
>>> I would say that the next term is 157. Indeed, the number of simplicial
>>> complexes with n vertices is (starting at n=0):
>>> 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
>>> It is in the OEIS... and well hidden in it. It is A006602, see comment
>>> by... Vladeta Jovovic. Since the zeroth term differs, it probably
>> deserves
>>> its own entry.
>>> Then, to count the connected ones, there is the usual summation over
>>> partitions:
>>> a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots} \prod_{i=1}^p
>>> a_{conn}(n_i)
>>> so the connected version is given by
>>> 0, 1, 1, 3, 14, 157, ...
>>> and the next two terms can be easily computed.
>>> Note that a_{conn}(0)=0 since the empty complex/graph/space is not
>>> connected.
>>>
>>> Best regards,
>>> Benoît
>>>
>>>
>>> On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
>> wrote:
>>>> Dear Seq Fans, Back in 1983 the physicist Greg Huber
>>>> sent me the initial terms of four sequences that arise in
>>>> the enumeration of simplicial complexes. The second one
>>>> is now A048143, and you can see an annotated
>>>> and corrected scan of his letter there.
>>>> But what about the first one? This is the number of
>>>> unlabeled connected simplicial complexes on n nodes. It
>>>> begins 1,1,3,14. It seems like a fundamental sequence in geometry.
>>>> If someone could find one or two more terms, we could add it to the
>> OEIS.
>>>> Vladeta Jovovic, where are you when we need you?
>>>>
>>>> Greg also mentions two other sequences arising from his study
>>>> of the early universe, but their definitions are not very explicit.
>>>>
>>>> _______________________________________________
>>>>
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
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