A new method to introduce the Möbius function and the Möbius transform

1994 ◽  
Vol 35 (6) ◽  
pp. 3099-3108 ◽  
Author(s):  
Nan‐xian Chen
Keyword(s):  
Möbius Function ◽  
New Method ◽  
Möbius Transform ◽  
Mobius Function

scholarly journals Factorization of the Characteristic Polynomial

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Joshua Hallam ◽  
Bruce Sagan
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Finite Lattice ◽  
Möbius Function ◽  
New Method ◽  
Mobius Function ◽  
International Audience

International audience We introduce a new method for showing that the roots of the characteristic polynomial of a finite lattice are all nonnegative integers. Our method gives two simple conditions under which the characteristic polynomial factors. We will see that Stanley's Supersolvability Theorem is a corollary of this result. We can also use this method to demonstrate a new result in graph theory and give new proofs of some classic results concerning the Möbius function.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

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