Some remarks on sign-balanced and maj-balanced posets (Q2483918)

If \(P\) is an \(n\)-element poset and \(\omega: P\to[n]= \{1,2,\dots,n\}\) is a bijection (labeling) of \(P\), then \((P,\omega)\) is a labeled poset. A linear extension \(f: P\to[n]\) is an order-preserving bijection which defines \(\pi= \pi(f)\in S_n\) by \(\pi(i)= j\) if \(f(\omega^{-1}(j))= i\), written as \(A_1A_2\cdots A_n\), \(\pi(i)= A_i\). Thus \(\pi\) is even or odd as a permutation and hence \(f\) is even or odd as the corresponding linear extension. \((P,\omega)\) is sign-balanced if it contains the same number of even as odd permutations (independently of \(\omega\)). The property of being sign-balanced is common to many (well-behaved) classes of posets as noted. In the general scheme of things bijections on the set of linear extensions which reverse purity imply sign-balance and are thus useful tools to be obtained directly or indirectly (the promotion and evacuation operators for example). The theory of partitions as exploited by the author (both past and present) and others who are known to practice generator-function-magic applied to standard domino tableaux (SDT's) as well as other configurations (SYT's, hook length posets) tends in the indirect direction. Shifting to the major-index (of old), \((P,\omega)\) is maj-balanced if the number of linear extensions with even major-index equals the number of those with odd major-index, with results (new) looked for related to (or induced by) sign-balanced facts and obtained (absence of \(P\)-domino tableaux for \((P,\omega))\), as well. A bouillabaisse with plenty of items and observations contained in a surrounding broth of recognizable flavor by a well-know chef operating in his favorite district.