
TL;DR
This paper introduces a M"{o}bius conjugation for functions on posets, leading to convolution identities and reciprocity theorems for hyperplane arrangements and Tutte polynomials, unifying various combinatorial results.
Contribution
It develops a new M"{o}bius conjugation framework for poset functions, deriving convolution identities and reciprocity theorems for arrangements and matroids.
Findings
Derived convolution identities for hyperplane arrangements.
Established reciprocity theorems for characteristic polynomials.
Unified known convolution identities for Tutte polynomials.
Abstract
Let be a locally finite poset with the interval space , and a ring with identity. We shall introduce the M\"{o}bius conjugation sending each function to an incidence function such that . Taking to be the intersection poset of a hyperplane arrangement , we shall obtain a convolution identity for the number of regions and the number of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial , which also leads to a combinatorial interpretation to the values for large primes . Moreover, all known convolution identities on Tutte polynomials of matroids will be direct consequences after specializing the poset and functions .
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