Few words added at Catalan number.

]]>I had occasion to create minimal entries for basic combinatorial concepts: *factorial*, *multinomial coefficient*

needed to be able to point to *connected graph*, so I created some minimum

I have given *multigraph* its own little entry, so that I can point to it (from discussion of Feynman diagrams).

I have given *finite graph* its own minimum entry, just so as to be able to point to it

Created a new page generalized graph based on the definition given in Hackney, Robertson and Yau’s recent book, which appears to be influenced at least in part by the paper of Kock cited on the page. As far as I can tell none of the other things on the nlab (e.g. quiver or graph or their associated sub-entries about pseudographs and so forth) deal with the case of the “exceptional cell.” If the notion I describe on this page is already somewhere on the nlab, I’d be happy to know that and get rid of the page I made.

]]>New entry combinatorial design

]]>I recently came across this concept yet have not seen a CT treatment of it. Its clearly set based but the “up to a permutation”, i.e. up to an isomorphism, caught my categorical eye. At worst, it seems like a good sandbox to develop a set intuition, IMO the first rung needed to climb the CT ladder. At best, I don’t know.

Any guidance on this topic and how it fits into CT would be most welcome.

“In combinatorics, the twelvefold way is a name given to a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. “

https://en.wikipedia.org/wiki/Twelvefold_way

]]>Few references collected as a start of entry spectrum of a graph redirecting also Ihara zeta function, prompted by today’s remarkable paper by Huang and Yau and thereby revived memory of a colloqium talk by Bass in which I enjoyed at University of Wisconsin in late 1990s.

]]>New references at symmetric function and new stub noncommutative symmetric function. An (unfinished?) discussion query from symmetric function moved here:

]]>David Corfield: Why does Hazewinkel in his description of the construction of $\Lambda$ on p. 129 of this use a graded projective limit construction in terms of projections of polynomial rings?

John Baez: Hmm, it sounds like you’re telling me that there are ’projections’

$\Lambda_{n+1} \to \Lambda_n$given by setting the $(n+1)$st variable to zero, and that Hazewinkel defines $\Lambda$ to be the limit (= projective limit)

$\cdots \to \Lambda_2 \to \Lambda_1 \to \Lambda_0$rather than the colimit

$\Lambda_0 \to \Lambda_1 \to \Lambda_2 \to \cdots$Right now I don’t understand the difference between these two constructions well enough to tell which one is ’right’. Can someone explain the difference? Presumably there’s more stuff in the limit than the colimit.

Mike Shulman: I think the difference is that the limit contains “polynomials” with infinitely many terms, and the colimit doesn’t. That’s often the way of these things.

Actually, on second glance, I don’t understand the description of the maps in the colimit system; are you sure they actually exist? What exactly does it mean to “add in new terms with the new variable to make the result symmetric”?

David Corfield: The two constructions are explained very well in section 2.1 of the Wikipedia article.

Mike Shulman: Thanks! Here’s what I get from the Wikipedia article: the projections are easy to define. They are surjective and turn out to have sections (as ring homomorphisms). The ring of symmetric functions can be defined either as the colimit of the sections, or as the the limit of the projections

in the category of graded rings. The limit in the category of all rings would contain too much stuff.

Lagrange inversion, redirecting also Lagrange inversion formula and Lagrange inversion theorem, previously wanted at Lambert W-function, noncommutative symmetric function and at Faà di Bruno formula.

]]>Annales de l’Institut Henri Poincaré D: Combinatorics, Physics, and their Interaction ]]>

New entry combinatorial Hopf algebra. Reference additions or updates in Hopf algebra, BV formalism, Hall algebra, graph homology, Marcelo Aguiar, renormalization.

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