Explicit enumeration of 321, hexagon-avoiding permutations

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The 321, hexagon-avoiding ( 321-hex) permutations were introduced and studied by Billey and Warrington (J. Alg. Comb. 13 (2001) 111–136). as a class of elements of S n whose Kazhdan–Lusztig polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensted's 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations.