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The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$

Statement

For a locally finite poset $P$ and $x \leq y$ ,

$μ_{P} (x, x) = 1,$

and, when $x < y$ ,

$\sum_{x \leq z \leq y} μ_{P} (x, z) = 0, \sum_{x \leq z \leq y} μ_{P} (z, y) = 0.$

Equivalently, off the diagonal,

$μ_{P} (x, y) = - \sum_{x \leq z < y} μ_{P} (x, z) = - \sum_{x < z \leq y} μ_{P} (z, y) .$

Either recurrence together with the diagonal values uniquely determines $μ_{P}$ interval by interval.

Facts & Assumptions

Given: A locally finite poset $P$ and comparable elements $x \leq y$ .

[F1]

$μ_{P} * ζ = δ = ζ * μ_{P}$ (The integer-valued Möbius function $μ_{P}$ of a locally finite poset).

[F2]

Convolution is the finite interval sum, $ζ$ is constantly $1$ , and $δ$ is $1$ on the diagonal and $0$ off it (The incidence functions $I (P, R)$ 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

technique · direct

1.1

Evaluating either inverse equation at $(x, x)$ gives $μ_{P} (x, x) = 1$ .

F1F2

1.2

Evaluating $μ_{P} * ζ = δ$ at $x < y$ gives $\sum_{x \leq z \leq y} μ_{P} (x, z) = 0$ .

F1F2

1.3

Evaluating $ζ * μ_{P} = δ$ at $x < y$ gives $\sum_{x \leq z \leq y} μ_{P} (z, y) = 0$ .

F1F2

2.1

Isolating the term $z = y$ in step 1.2 and the term $z = x$ in step 1.3 yields the two displayed recursive formulas.

step 1.2step 1.3algebra

3.1

Each right-hand side uses only proper subintervals, so induction on the finite cardinality of $[x, y]$ shows that either recurrence and the diagonal clause determine at most one function.

step 1.1step 2.1

4.1

Steps 1.1 through 3.1 prove both sums, both recurrences and uniqueness.

step 1.1step 1.2step 1.3step 2.1step 3.1∎

Depends on

Used by

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Direct dependencies and their dependencies through the next three levels: 62 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)

The Möbius recurrence: μP(x,x)=1μP​(x,x)=1 and both interval sums of μPμP​ vanish when x<yx<y

Statement

Facts & Assumptions

Proof

Depends on

Used by

Dependency tree · next 3 levels

Sources

The Möbius recurrence: $μ_{P} (x, x) = 1$ and both interval sums of $μ_{P}$ vanish when $x < y$