Classifying unavoidable Tverberg partitions Let T(d,r)  (r1)(d1)1 be the parameter in Tverbergu0027s theorem, and call a partition mathcal I of 1,2,ldots,T(d,r) into r parts a  Tverberg type . We say that mathcal I o ccurs xc2xa0in an ordered point sequence P if P contains a subsequence Pu0027 of T(d,r) points such that the partition of Pu0027 that is orderisomorphic to mathcal I is a Tverberg partition. We say that mathcal I is  unavoidable xc2xa0if it occurs in every sufficiently long point sequence.  In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for dle 4. Along the way, we study the avoidability of many other geometric predicates.  Our techniques also yield a large family of T(d,r)point sets for which the number of Tverberg partitions is exactly (r1)!d. This lends further support for Sierksmau0027s conjecture on the number of Tverberg partitions.

Subsequence containment by involutions Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in cal S_n which contain a given permutation tauincal S_k as a subsequence; this number depends on the patterns of the first j values of tau for 1leq jleq k. We then use this to define a partition of cal S_k, analogous to Wilfclasses in the study of pattern avoidance, and examine properties of this equivalence.  In the process, we show that a permutation tau_1ldotstau_k is layered iff, for 1leq jleq k, the pattern of tau_1ldotstau_j is an involution.  We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size n which contain a fixed tableau of size k as a subtableau.