#14542: Implement arithmetic product of cycle index series
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Reporter: agd | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.10
Component: combinatorics | Resolution:
Keywords: species, cycle index | Work issues:
Report Upstream: N/A | Reviewers:
Authors: | Merged in:
Dependencies: | Stopgaps:
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Comment (by darij):
line 602: "g" should be inside double backticks.
line 620: Where does the "1 +" come from? Is there a regular octopus on 0
vertices? (I don't think so.)
line 623: xrange will be deprecated in Python 3.0; just saying.
line 643: Please explain what the arithmetic product of partition is
supposed to mean. The Maia-Mendez paper defines the arithmetic product of
polynomials; as far as I understand, by the arithmetic product of two
partitions \lambda and \mu you mean the partition obtained by writing down
gcd(\lambda_i, \mu_j) times the integer lcm(\lambda_i, \mu_j) for every i
and every j, and then sorting the resulting sequence in nonincreasing
order. That's fine, but you should clarify the relation between
- this operation on partitions,
- the Maia-Mendez operation on polynomials, and
- the operation on symmetric functions that you have actually implemented.
(You are interpreting the x_i of Maia-Mendez as the i-th power sum, as far
as I understand, and the partition \lambda corresponds to the product
p_\lambda = p_{\lambda_1} p_{\lambda_2} ....)
About your implementation of {{{arith_prod_of_partitions}}}: I'm not sure
how fast it is. It computes each lcm and each gcd very often (maxval
times), whereas the definition I gave would only have to compute it once.
On the other hand, the definition I gave requires sorting (or, better,
insort).
line 685: I disagree with this, as on line 620.
line 690: Why do you put a p.zero() after res? (I'm new to the
CycleIndexSeries class, so maybe it's important...)
On another note rather unrelated to Sage: The species viewpoint on the
arithmetic product shows that the arithmetic product on the ring of
symmetric functions (the one given by
p_\lambda \boxdot p_\mu = \prod_{i,j} p_{\lcm(\lambda_i,
\lambda_j)}^{\gcd(\lambda_i, \lambda_j)}
for all partitions \lambda and \mu) is defined over the integers, not just
over the rationals (despite the p_\lambda not forming a Z-basis of Symm).
Is there an algebraic proof of this? It looks like a nice application of
species to proving a nontrivial algebraic result. Is there a species-
theoretical proof of the integrality of \Delta_3 in
http://mathoverflow.net/questions/120924/is-the-renormalized-third-
comultiplication-on-mathbfsymm-integral as well?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14542#comment:4>
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