  In this note we give a(nother) combinatorial proof of an old result of
Baik--Rains: that for appropriately considered independent geometric weights,
the generating series for last passage percolation polymers in a $2n \times n
\times n$ quarter square (point-to-half-line-reflected geometry) splits as the
product of two simpler generating series---that for last passage percolation
polymers in a point-to-line geometry and that for last passage percolation in a
point-to-point-reflected (half-space) geometry, the latter both in an $n \times
n \times n$ triangle. As a corollary, for iid geometric random variables---of
parameter $q$ off-diagonal and parameter $\sqrt{q}$ on the diagonal---we see
that the last passage percolation time in said quarter square obeys
Tracy--Widom $\mathrm{GOE}^2$ fluctuations in the large $n$ limit as both the
point-to-line and the point-to-point-reflected geometries have known GOE
fluctuations. This is a discrete analogue of a celebrated Baik--Rains theorem
(the limit $q \to 0$) and more recently of results from Bisi's PhD thesis (the
limit $q \to 1$).
