Prove combinatorial identities by counting incidences in finite bipartite relations.
npx skills add https://github.com/bneb/perqed --skill double-counting
Please help me install this Agent Skill. Skill: double_counting Source: https://github.com/bneb/perqed/tree/main/.agents/skills/double_counting Command: npx skills add https://github.com/bneb/perqed --skill double-counting
Double counting (also: counting in two ways, Fubini for finite sets) establishes an algebraic identity by considering a bipartite relationship R ⊆ A × B. Counting the total number of "incidences" |R| by summing over the A-side gives one expression; summing over the B-side gives another. Equating them yields the identity.
Provide a double-counting proof of a chosen identity by counting incidences in a finite bipartite relation.
High-intent search queries and answers about installing and using this skill.
Double counting proves combinatorial identities by counting incidences in a finite bipartite relation R ⊆ A × B. Summing over the A-side and B-side yields two expressions for the total count, and equating them establishes the identity.
Prove the handshaking lemma and Vandermonde convolution in Lean by structuring the proof around Finset-like sums and incidence counts. This Skill formalizes the bipartite relation decomposition to equate sums over both sides.
Counting in two ways requires basic set theory and finite counts. You need to understand finite sets, bipartite relations R ⊆ A × B, and how to compute Finset-like sums over both sides to apply the technique correctly.
The best way to equate sums over finite sets for binomial identities is double counting. Define a finite bipartite relation, count total incidences by summing over each side separately, and equate the resulting expressions to derive the identity.
You should not use double counting for finite set proofs when your problem cannot be modeled as a finite bipartite relation R ⊆ A × B. The technique requires that both sides admit natural decompositions yielding meaningful Finset-like incidence sums.
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