The Generalized Interval Polynomial of a Poset

TL;DR

The paper introduces the generalized interval polynomial $\Phi(P; s, t, x, y, z)$ for finite posets, encoding counts of upsets, downsets, and their intersection via the sizes of generating antichains. It develops the algebraic properties of $\Phi$, including its behavior under duality and standard poset operations, and demonstrates that $\Phi$ distinguishes series-parallel posets by reconstructing composition trees from its irreducible factors. The authors show that $\Phi$ does not determine a poset up to isomorphism in general, but provides explicit computations for important poset families and rules for deriving combinatorial information from the polynomial. Connecting to causal set theory, they discuss how interval distributions in causal sets relate to spacetime dimension and propose that generalized interval data may enhance dimension estimation. Overall, the work provides a rich generating-function framework for poset analysis with potential applications in discrete quantum gravity.

Abstract

For any finite poset we define a generating polynomial counting upsets, downsets, and their intersection. We investigate the behaviour of this polynomial with respect to poset operations, show that it distinguishes series-parallel posets, and comment on connections to the causal set approach to quantum gravity.

Figures (3)

• Figure 1: Some example posets of interest.
• Figure 2: An SP poset and its composition tree.
• Figure 3: Two SP posets that are not distinguished by the ordinary interval polynomial.

Theorems & Definitions (23)

• Definition 1
• Example 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Example 2
• Example 3