9,927 reputation
23178
bio website math.ucla.edu/~pak
location Los Angeles, CA
age
visits member for 5 years, 6 months
seen 1 hour ago

Aug
24
comment Important formulas in Combinatorics
Um, we gave a truly bijective proof of MMT in this paper, section 2. Sorry for the self-promotion. arxiv.org/abs/math/0607737
Aug
20
comment Important formulas in Combinatorics
Sam, you are right. Sometimes the names are misleading...
Aug
19
comment Important formulas in Combinatorics
I think we agree on everything except "what is a formula". It's not really a formal discussion, but if this is a formula, then so is every theorem in combinatorics.
Aug
18
comment Important formulas in Combinatorics
Yeah, well - this identity is truly classical and extremely important in Partition Theory (see also Hardy & Wright Number Theory book on this). Your statement is sort of like saying "matrices are important especially because SU(3) helps explain quarks". True, but a minority view.
Aug
18
comment Important formulas in Combinatorics
Um, no. They were originally defined and studied as multiplicity constants in the tensor product of two GL(n,C) modules. Those are non-negative integers by definition. A combinatorial interpretation came much later indeed, but that speaks to the importance (which I agree with completely), not whether this is a formula akin the HLF.
Aug
18
comment Important formulas in Combinatorics
That's not a formula in a sense that it does not allow computing LR coeff. faster than via the other standard definitions.
Aug
18
comment Important formulas in Combinatorics
The Jacobi triple product identity was proved bijectively by Sylvester (see my partition bijections survey). Reviewing the literature I found about 11 proofs by others, all equivalent but phrased differently. Your answer suggests that "particle sea" proof is somehow different and modern. It is in fact equivalent to the original Sylvester's bijection.
Aug
4
awarded  Good Answer
Jul
24
revised What are some examples of interesting uses of the theory of combinatorial species?
link updated
May
28
answered Complexity of planar scissor congruence
May
17
awarded  Good Answer
May
10
revised Random Alternating Permutations
update
May
2
awarded  Autobiographer
Apr
27
awarded  Good Question
Apr
21
awarded  Promoter
Apr
15
awarded  Curious
Apr
14
comment 3-colorings of the unit distance graph of $\Bbb R^3$
@JMPerry - Yes. Every triangle in $\Gamma$ has all sides unit length.
Apr
14
comment 3-colorings of the unit distance graph of $\Bbb R^3$
@JMPerry. I don't understand your question/objection. Please write it in full if possible.
Apr
14
comment 3-colorings of the unit distance graph of $\Bbb R^3$
@JMPerry - No, since $2\beta \neq 1$. I meant to write "unit length triangle" in my comment above, which would mean it's a triangle in $\Gamma$.
Apr
14
revised 3-colorings of the unit distance graph of $\Bbb R^3$
added 12 characters in body