Linked Questions
18
votes
2
answers
1k
views
A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
- 1,190
19
votes
3
answers
2k
views
Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
- 10.5k
2
votes
2
answers
2k
views
Inversion of Laurent series
For a power series $f(z) = \sum_{i=0}^{\infty} a_i z^i$ with $a_1$ nonzero, Lagrange's inversion formula gives an explicit way to compute the Taylor coefficients of the inverse function.
Is there any ...
- 21k
9
votes
1
answer
889
views
Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 10.5k
7
votes
0
answers
579
views
Guises of the noncrossing partitions (NCPs)
From "Noncrossing partitions in surprising locations" by Jon McCammond:
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
- 10.5k
3
votes
1
answer
159
views
Proving that two sequences of polynomials defined over partitions are inverse to each other
For any fixed $c>0$ consider the polynomials
\begin{align*}
& p_n(X_1,X_2,\ldots) := \frac{n!}{c} \sum\limits_{b=1}^n \frac{c^b}{b!(n+1-b)!} \sum\limits_{\substack{l_1,\ldots,l_b \geq 1 \\ ...
- 1,295
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
3
votes
0
answers
181
views
Refined f- and h-partition polynomials of the associahedra
The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra ...
- 10.5k
1
vote
1
answer
177
views
Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond
Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
- 10.5k
3
votes
0
answers
116
views
A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
3
votes
0
answers
65
views
A combinatorial Dyson-Schwinger equation, tree diagrams, and compositional inversion of a Laurent series
In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9:
Combinatorial Dyson-Schwinger equations are functional equations with solutions in
$...
- 10.5k