Abstract
Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of $s$-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance.
Highlights
A permutation w is called vexillary if and only if it avoids the patterns [2 1 4 3], i.e., there are no indices a < b < c < d such that w(b) < w(a) < w(d) < w(c)
Vexillary permutations were defined by Lascoux and Schutzenberger [9] in the 1980s
In the 1990s, Macdonald [10] and Fulton [8] gave a characterizations for the vexillary permutations in terms of the essential sets
Summary
A permutation w is called vexillary if and only if it avoids the patterns [2 1 4 3], i.e., there are no indices a < b < c < d such that w(b) < w(a) < w(d) < w(c). Anderson and Fulton [2, 4] provided a different characterization for vexillary signed permutations They defined them through a specific triple of integers: given three s-tuple of positive integers τ = (k, p, q), where k = (0 < k1 < · · · < ks), p = (p1 · · · ps > 0), and q = (q1 · · · qs > 0), satisfying pi − pi+1 + qi − qi+1 > ki+1 − ki for 1 i s − 1, one constructs a signed permutation w = w(τ ). Anderson and Fulton characterize vexillary signed permutations in terms of essential sets, pattern avoidance, and Stanley symmetric functions [4]. Considering the pattern avoidance criterion, the set of theta-vexillary signed permutations form a new class of permutations according to the “Database of Permutation Pattern Avoidance” maintained by Tenner [11]
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