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{\bf Roger E. Behrend}
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{\bf Osculating Paths and Oscillating Tableaux}
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The combinatorics of certain tuples of osculating lattice paths is
studied, and a relationship with oscillating tableaux is obtained.
The paths being considered have fixed start and end points on
respectively the lower and right boundaries of a rectangle in the
square lattice, each path can take only unit steps rightwards or
upwards, and two different paths within a tuple are permitted to share
lattice points, but not to cross or share lattice edges. Such path
tuples correspond to configurations of the six-vertex model of
statistical mechanics with appropriate boundary conditions, and they
include cases which correspond to alternating sign matrices. Of
primary interest here are path tuples with a fixed number~$l$ of
vacancies and osculations, where vacancies or osculations are points
of the rectangle through which respectively no or two paths pass. It
is shown that there exist natural bijections which map each such path
tuple~$P$ to a pair~$(t,\eta)$, where~$\eta$ is an oscillating tableau
of length~$l$ (i.e., a sequence of~$l+1$ partitions, starting with
the empty partition, in which the Young diagrams of successive
partitions differ by a single square), and~$t$ is a certain,
compatible sequence of~$l$ weakly increasing positive integers.
Furthermore, each vacancy or osculation of~$P$ corresponds to a
partition in~$\eta$ whose Young diagram is obtained from that of its
predecessor by respectively the addition or deletion of a square.
These bijections lead to enumeration formulae for tuples of osculating
paths involving sums over oscillating tableaux.
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