Stanley, Richard P. An equivalence relation on the symmetric group and multiplicity-free flag \(h\)-vectors. (English) Zbl 1291.05012 J. Comb. 3, No. 3, 277-298 (2012). Summary: We consider the equivalence relation on the symmetric group \(S_n\) generated by the interchange of two adjacent elements \(a_i\) and \(a_{i+1}\) of \(w=a_1\cdots a_n \in \mathcal{S}_n\) such that \(|a_{i} - a_{i+1}|=1\). We count the number of equivalence classes and the sizes of equivalence classes. The results are generalized to permutations of multisets using umbral techniques. In the original problem, the equivalence class containing the identity permutation is the set of linear extensions of a certain poset. Further investigation yields a characterization of all finite graded posets whose flag \(h\)-vector takes on only the values \(0,\pm 1\). Cited in 2 ReviewsCited in 9 Documents MSC: 05A15 Exact enumeration problems, generating functions 06A07 Combinatorics of partially ordered sets Keywords:symmetric group, linear extension, flag \(h\)-vector PDFBibTeX XMLCite \textit{R. P. Stanley}, J. Comb. 3, No. 3, 277--298 (2012; Zbl 1291.05012) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: From applying the ”rational mean” to the number e. a(0)=1; a(n) = 2*3^(n-1) for n >= 1. Number of free generators of degree n of symmetric polynomials in 4 noncommuting variables. Expansion of x^2*(1-x^2)*(1-3*x^2)/(1-x-5*x^2+4*x^3+5*x^4-3*x^5).