Möbius Polynomials

Murray, Will

doi:10.4169/math.mag.85.5.376

Cited by 2 publications

(4 citation statements)

References 6 publications

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“…We then compute these polynomials for certain families of matroids and find special roots. Of particular interest is that the generalized Möbius function has −1 as a root for modular matroids, which mimics Theorem 1 in [21]. However, we show that the converse is not true and so one is led to question what do these polynomial count?…”

Section: Accepted Manuscriptmentioning

confidence: 64%

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A non-associative incidence near-ring with a generalized Möbius function

Johnson

Wakefield

2023

Ars Math. Contemp.

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Section: Accepted Manuscriptmentioning

confidence: 64%

“…As an application we build two different polynomials from the J-function: a generalized characteristic polynomial and a generalized Möbius polynomial (see [17] and [21] for Möbius polynomials). It turns out that these polynomials have some interesting properties that are not apparent from the surface.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

A non-associative incidence near-ring with a generalized Möbius function

Johnson

Wakefield

2023

Ars Math. Contemp.

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“…As a kind of application we build two different polynomials from the J-function: a generalized characteristic polynomial and a generalized Möbius polynomial (see [13] and [17] for Möbius polynomials). It turns out that these polynomials have some interesting properties that are not apparent from the surface.…”

Section: Introductionmentioning

confidence: 99%

“…Then we compute these polynomials for certain families of matroids and find special roots. Of particular interest is that the generalized Möbius function has -1 as a root for modular matroids which mimics Theorem 1 in [17]. However, the converse is not true and so one is led to question what do these polynomial count?…”

Section: Introductionmentioning

confidence: 99%

A non-associative incidence near-ring with a generalized Möbius function

Johnson¹,

Wakefield²

2022

Preprint

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There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the Möbius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbius function we define analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized Möbius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.

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Möbius Polynomials

Abstract: Benjamin and Robert Mena for suggesting this investigation and many useful subsequent thoughts. The referees also offered many helpful ideas that dramatically improved this paper.

Cited by 2 publications

References 6 publications

A non-associative incidence near-ring with a generalized Möbius function

A non-associative incidence near-ring with a generalized Möbius function

A non-associative incidence near-ring with a generalized Möbius function

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