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The Möbius recurrence: and both interval sums of vanish when
Statement
For a locally finite poset and ,
and, when ,
Equivalently, off the diagonal,
Either recurrence together with the diagonal values uniquely determines interval by interval.
Facts & Assumptions
Given: A locally finite poset and comparable elements .
Convolution is the finite interval sum, is constantly , and is on the diagonal and off it (The incidence functions of a locally finite poset and their convolution, The delta and zeta incidence functions, A finite sum in a commutative monoid indexed by an arbitrary finite set).
Proof
Evaluating either inverse equation at gives .
Evaluating at gives .
Evaluating at gives .
Isolating the term in step 1.2 and the term in step 1.3 yields the two displayed recursive formulas.
Each right-hand side uses only proper subintervals, so induction on the finite cardinality of shows that either recurrence and the diagonal clause determine at most one function.
Steps 1.1 through 3.1 prove both sums, both recurrences and uniqueness.
Depends on
Used by
- The number-theoretic Möbius function is multiplicative on coprime positive integers Corollary
- A parabolic quotient interval of S4 whose Möbius value is 0, so the Eulerian sign formula does not extend to quotients Counterexample
- Deleted-position labels from a fixed reduced expression, the lexicographic shelling criterion, and Möbius data Definition
- A rank-three chain labeling translated into facets of the order complex Example
- The endpoint Möbius value of the four-element diamond is 1 Example
- The full Möbius table of the Boolean lattice 2^[3] Example
- The Möbius function on the divisor poset of 12 and its agreement with μ(1),μ(2),μ(3),μ(4),μ(6),μ(12) Example
- The Möbius table of a four-element chain Example
- The Möbius value of the rank-three interval [e,c] in S4 from the recurrence, with the parity and falling-chain checks Example
- Lexicographic chain shelling and the falling-chain Möbius formula Lemma
- Bruhat intervals are Eulerian: parity balance of the elements, and the Möbius function of a full interval Theorem
- For A⊆ B in a finite Boolean lattice, μ(A,B)=(-1)^| B∖ A| Theorem
- Möbius inversion on a lower-finite poset, with the dual upper-finite form Theorem
- On a finite chain, the Möbius function is 1 on the diagonal, -1 on covers and 0 on longer intervals Theorem
- The Möbius function of a product poset is the product of the Möbius functions Theorem
- The number-theoretic Möbius function is the poset Möbius function of divisibility: μ(n)=μ_∣(1,n) Theorem
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- F. Gotti, Incidence Algebras, MIT 18.211 notes (standard reference, not scraped)
- R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)