Last passage percolation (LPP) in an $n\times n$ lower triangular domain has nice connections with various generalizations of Schur measures. LPP along an anti-diagonal, from $(1,n)$ to $(n,1)$, gives a distribution of a highest column of a random composition with respect to a Demazure measure (a non-symmetric analog of a Schur measure). LPP along a main diagonal, from $(1,1)$ to $(n,n)$, is distributed as a marginal of a Pfaffian Schur process. In the first case we show that the asymptotics for the constant specialization is governed by the GOE Tracy-Widom distribution, in the second case - by the GSE Tracy-Widom distribution. In the latter case we were also able to study the truncated lower triangular case, obtaining an interesting generalization of the GSE Tracy-Widom distribution.