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Results tagged with co.combinatorics
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user 10881
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
4
votes
A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?
Sagemath can do that too
sage: x = PowerSeriesRing(QQ,'x').gen()
sage: f = sum(x**n/(n+1)**2 for n in range(20)).O(20)
sage: f.jacobi_continued_fraction()
((-1/4, -7/144),
(-13/28, -647/110 …
- 3,597
7
votes
2
answers
258
views
About the sum of rectangular power sums
Let $n \geq 1$ be an integer and consider the symmetric function
$$D_n = \sum_{d|n} p_d^{n/d},$$
where $p_{d}$ are the power-sum symmetric functions.
It can be checked up to $n=35$ that the symmetric …
- 3,597
9
votes
Accepted
Software for recognizing algebraic or D-finite formal power series
Fricas is good at that. It can be accessed via sage, once installed.
sage: L=[catalan_number(i) for i in range(20)]
sage: fricas.guessHolo(L) …
- 3,597
3
votes
0
answers
120
views
About finite posets without intervals of size 3
Let $P$ be a finite poset (partially ordered set).
I am wondering whether the following condition on $P$ has been studied somewhere:
(#) No interval $[a,b]$ in $P$ has $3$ elements.
Note that interv …
- 3,597
3
votes
Grassmannian cluster algebra of infinite type has no trees in its mutation class
Regarding question Q2, one can go a little bit further and describe simple diagrams with few edges for some more cases.
Let us talk about $Gr(p, p+q)$, so that there is a symmetry between $p$ and $q$. …
- 3,597
5
votes
Accepted
an algebra generated by some known series
This has been considered by Dimitri Zvonkine, see his article "An algebra of power series arising in the
intersection theory of moduli spaces of curves
and in the enumeration of ramified coverings
of …
- 3,597
2
votes
Accepted
Birkhoff Lattice of a forest
This set of lattices is the closure of the set {poset with one element} under two allowed operations: adding a top element or taking a Cartesian product. This implies that the Möbius numbers are in $\ …
- 3,597
6
votes
0
answers
276
views
universality for large deviations?
This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large …
- 3,597
15
votes
0
answers
588
views
On some special spanning trees of grid graphs
I would like to know if there are existing results on the following objects:
spanning trees of a grid graph, with no corridor
where a corridor is a vertex having exactly two neighbors, on opposi …
- 3,597
4
votes
Combinatorial interpretation of composition of power series?
When rephrased (using the positive generating series $g=-f$) as the identity $g(-g(-t))=t$, this suggests that there may exist a quadratic nonsymmetric operad with this generating series, which is Kos …
- 3,597
1
vote
Flow of an integer
As a service to the community, here are these digraphs in sage:
def divisor_graph(n):
"""
Mathoverflow 159319
"""
vert = divisors(n)
return DiGraph([(a, b, b / a) for b in vert
…
- 3,597
2
votes
Number of trees with the same matching number
This can be done using the canonical coloring of vertices of trees into 3 colors that can be found in
J. Zito, "The structure and maximum number of maximum independent sets in trees"
S. Coulomb and …
- 3,597
2
votes
Kahler differentials on cluster varieties
There is not much known in general, as far as I know.
There is a nice 2-form (called the Weil-Petersson 2-form) defined using the cluster algebra structure.
This can be found in article "The Weil-Pe …
- 3,597
7
votes
Symmetric powers of Schur polynomials
this could be done in sage:
sage: B3 = WeylCharacterRing("B3", style="coroots")
sage: spin = B3(0,0,1)
sage: spin.symmetric_power(6)
B3(0,0,0) + B3(0,0,2) + B3(0,0,4) + B3(0,0,6)
sage: A3 = WeylChar …
- 3,597
4
votes
Temperley-Lieb algebras for other Weyl groups?
There are some ad-hoc definitions for some types. Type B can be defined using diagrams that have a left-right symmetry. Tammo tom Dieck has proposed a definition for type D here: (http://www.uni-math. …
- 3,597