Logarithmic Tree Factorials

To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargavas factorials to a natural combinatorial setting suitable for studying questions around generalized factorials. We discuss several basic aspects of the framework in this paper. In particular, we relate the growth of the sequence of logarithmic factorials associated to a tree to the transience of the random walk and the existence of a harmonic measure on the tree, obtain an equidistribution theorem for factorial-determining-sequences of subsets of local fields, and provide a factorial-based characterization of the branching number of infinite trees. Our treatment is based on a local weighting process in the tree which gives an effective way of constructing the factorial sequence.

$lambda$-factorials of $n$

Recently, by the Riordans identity related to tree enumerations, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, end{eqnarray*} Sun and Xu derived another analogous one, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, end{eqnarray*} where $D_{k}$ is the number of permutations with no fixed points on ${1,2,dots, k}$. In the paper, we utilize the $lambda$-factorials of $n$, defined by Eriksen, Freij and W$ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and another two algebraic proofs. Using the umbral representation of our generalized identity and the Abels binomial formula, we deduce several properties for $lambda$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $mathsf{PSL}_2(mathbb{Z})$ on $mathsf{P}^1(mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

A formalism of abstract quantum field theory of summation of fat graphs

In this work we present a formalism of abstract quantum field theory for fat graphs and its realizations. This is a generalization of an earlier work for stable graphs. We define the abstract correlators $mathcal F_g^mu$, abstract free energy $mathcal F_g$, abstract partition function $mathcal Z$, and abstract $n$-point functions $mathcal W_{g,n}$ to be formal summations of fat graphs, and derive quadratic recursions using edge-contraction/vertex-splitting operators, including the abstract Virasoro constraints, an abstract cut-and-join type representation for $mathcal Z$, and a quadratic recursion for $mathcal W_{g,n}$ which resembles the Eynard-Orantin topological recursion. When considering the realization by the Hermitian one-matrix models, we obtain the Virasoro constraints, a cut-and-join representation for the partition function $Z_N^{text{Herm}}$ which proves that $Z_N^{text{Herm}}$ is a tau-function of KP hierarchy, a recursion for $n$-point functions which is known to be equivalent to the E-O recursion, and a Schrodinger type-equation which is equivalent to the quantum spectral curve. We conjecture that in general cases the realization of the quadratic recursion for $mathcal W_{g,n}$ is the E-O recursion, where the spectral curve and Bergmann kernel are constructed from realizations of $mathcal W_{0,1}$ and $mathcal W_{0,2}$ respectively using the framework of emergent geometry.

The Generating Function for the Dirichlet Series $L_m(s)$

The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by ${s_{m,n}}_{ngeq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)sec (btx)(pm cos ((b-p)tx)pm sin (ptx))$, where p is an integer satisfying $0leq pleq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.