Session Time & Room : 2D (Aug.22, 15:30-17:10) @G302
Type : Proposal of Minisymposium
Abstract :
To a finite poset $P$ we associate two algebraic objects: the order polynomial $\Omega(P,n)$, which counts the number of order preserving labeling maps of posets from $P$ to $1 < 2 < \ldots < n$; and the generating series $\sum \Omega(P,n)x^n$, called order series.
This algebraic setting leads to interactions with operad theory, metric spaces, cellular automata, and combinatorial species. In this mini-symposium, the experts will introduce those tools and expose their contributions as well as open questions. The talks will include theoretical results as well as applications to nonlinear signal-flow graphs.
00654 (1/1) : 2D @G302 [Chair: Eric Rubiel Dolores Cuenca]
[02151] The operad of finite posets acts on zeta values
Format : Talk at Waseda University
Author(s) :
Eric Dolores Cuenca (Yonsei University)
Abstract : Consider Riemann zeta function $\zeta(k)=\sum_{n=1}^\infty\frac{1}{n^k}$. We develop a theory in which the zeta values and the order polytopes share the same combinatorial information. More precisely, we show that the operad of finite posets acts on Stanley order polynomials, and the aforementioned operad acts on a set of numbers generated by zeta values.
To demonstrate that the action of the operad on zeta values is not trivial, we explicitly compute a quaternary associative non commutative operation.
[01777] What is -Q for a poset Q?
Format : Talk at Waseda University
Author(s) :
Masahiko Yoshinaga (Osaka University)
Abstract : In the context of combinatorial reciprocity, it is a natural question to ask what "−Q" is for a poset Q. In a previous work, the definition "−Q:=Q×R with lexicographic order" was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley's reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of Q is metrizable.
[04172] A new expression for the order polynomial
Format : Online Talk on Zoom
Author(s) :
Fengming Dong (Nanyang Technological University )
Abstract : In 1970s, Stanley introduced the order polynomial of a poset $P$. For a poset $P$, a mapping $\sigma: P\rightarrow [m]$ is said to be order-preserving if $u\preceq v$ implies that $\sigma(u)\le \sigma(v)$. The order polynomial $\Omega(P,x)$ is defined to be the function which counts the number of order-preserving mappings $\sigma:P\rightarrow [m]$ whenever $x=m$ is a positive integer. In this talk, I will introduce an expression for $\Omega(P,x)$, and an expression for the chromatic polynomial of a graph by applying this new result on order polynomial.
[02099] Polychrony as chinampas
Format : Online Talk on Zoom
Author(s) :
José Antonio Arciniega-Nevárez (Universidad de Guanajuato)
Abstract : In this talk, we will discuss the effect of stimulating some nodes at certain time (initial condition) in a signal flow path with auto-edges. The stimuli are applied at different times. The effect of these stimuli propagates through the vertices of the path, causing other nodes, which we will call secondary nodes, to cascade. We are interested in cascades in which the initial stimuli are less than or equal to the number of secondary vertices. This can be transferred to the study of a graph with weights, these weights being the times at which a signal travels from the output vertex to the arrival vertex. We are interested in knowing the time at which a node will be activated under a certain initial condition. The problem is nonlinear, so to deal with time, we propose to study a latiz, where, vertically, the time is described and horizontally, the vertices of the path are repeated. In this way, a vertex of the latiz will be connected to another one to which it was already connected in the path but at a different height, depending on the time.
If the time is always t=1, the problem becomes to study an automaton, where a cell at time t lives depending on the cells alive at time t-1, in particular we are interested in the rule 192 with the difference that the automaton is reactivated at different times (with the initial condition). The generated automata,that we call chinampas, have topological and combinatorial properties that allow us to solve the initial problem. Moreover, we can characterize automata in which the number of cells outside the initial condition are equal to or one more than those of the initial condition.
For the case in which we have more than one of these cells, we have translated the problem to one of triangular series with order conditions. These series can be obtained from partially ordered sets (posets). We have noted that the series obtained are known as generalizations of Stanley polynomials whose coefficients count the different ways of labeling a poset preserving the order.
The problem was inspired by the behavior of a neural network so this work is related to those who study such neural networks as polychrony groups, in fact, this is a particular case of polychrony.
