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9h
comment Even parking functions and spanning trees of complete bipartite graphs
I agree that something like this "should" work. Basically you are composing bijective proofs that the number is $m^{2m-2}$. But somehow I could hope for a more "tree-like" proof akin to known proofs for the non-bipartite case. For instance, we could require a bijection in the classical case that restricts to the bipartite case...
1d
comment How does subdividing an edge change the Tutte polynomial of graph at $x=0$?
Conjecture 1 is absolutely true. Just apply the spanning tree activity definition of the Tutte polynomial...
Feb
3
comment Even parking functions and spanning trees of complete bipartite graphs
The number of spanning trees of $K_{n,m}$ is $n^{m-1}m^{n-1}$ (oeis.org/A072590). Whereas the even parking functions described are (after dividing by two) $(2m-1,m)$-rational parking functions; it is known that the number of $(a,b)$-parking functions with $\mathrm{gcd}(a,b)=1$ is $b^{a-1}$. (The same classical proof of Pollak works to show this.) So to answer your question, no I don't know what it would look like for $K_{n,m}$.
Feb
3
asked Even parking functions and spanning trees of complete bipartite graphs
Feb
3
comment Concept of Facets in the structure of reductive algebraic groups
In polyhedral combinatorics the accepted terminology is that a "facet" is a "face" of maximal dimension. It seems quite strange to use facet to mean flats of different dimension.
Jan
25
revised q-Integer-valued polynomials
added 202 characters in body
Jan
17
comment Variants of Szemeredi's regularity lemma
However I cannot find that referenced paper. Perhaps it is still in preparation.
Jan
17
comment Variants of Szemeredi's regularity lemma
... Finally, it is further shown in [6] that whether or not an equitable partition is required has a negligible effect on M(epsilon)." [6] cites J Fox, A. Grinshpun, L. M. Lovász, and Y. Zhao, On regularity lemmas and their applications, In preparation.
Jan
17
comment Variants of Szemeredi's regularity lemma
I'll just quote that paper: "A vertex partition of a graph is equitable if any two parts differ in size by at most one. In the statement of the regularity lemma, it is often added that the vertex partition is equitable. There are several good reasons not to add this requirement to the regularity lemma. First, our main result, which gives a lower bound on M(epsilon) whose height is on the same order as the upper bound, does not need this requirement. Second, the proof of the upper bound is cleaner without it....
Jan
16
comment Variants of Szemeredi's regularity lemma
This paper of Fox and Lovász discusses a few different versions of the regularity lemma, and why it is preferable to state the lemma one way rather than another: arxiv.org/abs/1403.1768
Jan
12
comment Edge chromatic number of hypergraphs
Can you be a bit more precise about how the stated question is equivalent to the EFL conjecture?
Dec
24
awarded  Necromancer
Dec
22
comment Time Hierarchy Theorem and P vs NP
While this answer is no doubt correct (if $P \neq NP$ there are problems outside $P$ but not $NP$-hard), it is unclear if the OP means something more specific by "time class" than "complexity class." It is not clear to me that even $NP$ qualifies as a "time class." So I don't know that $NP$-intermediate automatically qualifies as a "time class" either.
Dec
20
comment Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?
Is there a simple proof of this fact using representation theory?
Dec
14
comment Applications of Representation Theory in Combinatorics
A general comment on this question is that many of the answer are not about combinatorial identities per se.
Dec
14
comment Applications of Representation Theory in Combinatorics
@AllenKnutson: In fact, I believe recent work of Watanabe does give precisely a representation-theoretical reason for Schubert structure constants on full flag manifolds to be nonnegative. See arxiv.org/abs/1410.7981.
Dec
13
comment $q$-connectedness of random digraphs obtained from a fixed graph
Very nice proof!
Dec
13
accepted $q$-connectedness of random digraphs obtained from a fixed graph
Dec
11
comment Which journals publish experimental results in pure maths?
Also, almost all journals will publish counterexamples to important conjectures, which in principle could have been found via computation.
Dec
11
comment The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?
It does not directly address the question, but it's probably worth linking to this blogpost of Terry Tao: terrytao.wordpress.com/2015/12/10/…