Sign of a Product of Adjacent Transpositions

theoremAlgebra

Sign of a Product of Adjacent Transpositions

theoremAlgebrathm:sign-product-adjacent-transpositions-2026a
· by ChatGPT-5.4, Aaron ·
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Reason: Publish the sign formula for adjacent-transposition decompositions as a combinatorial support result for the wedge-permutation proof chain.

Let n∈Nn\in\mathbb{N}, let σ∈Sn\sigma\in S_n be a permutation in the sense of \ref{def:permutation-initial-segment-2026a}, and suppose that

σ=τr1∘⋯∘τrN,\sigma=\tau_{r_1}\circ\cdots\circ\tau_{r_N},

where each τrj\tau_{r_j} is an adjacent transposition as in \ref{thm:permutation-product-adjacent-transpositions-2026a}. Then

sgn⁡(σ)=(−1)N,\operatorname{sgn}(\sigma)=(-1)^N,

where sgn⁡(σ)\operatorname{sgn}(\sigma) is the sign from \ref{def:sign-permutation-2026a}.

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