  The Schur $\mathsf{Lie}$-multiplier of Leibniz algebras is the Schur
multiplier of Leibniz algebras defined relative to the Liezation functor. In
this paper, we study upper bounds for the dimension of the Schur
$\mathsf{Lie}$-multiplier of $\mathsf{Lie}$-filiform Leibniz $n$-algebras and
the Schur $\mathsf{Lie}$-multiplier of its $\mathsf{Lie}$-central factor. The
upper bound obtained is associated to both the sequences of central binomial
coefficients and the sum of the numbers located in the rhombus part of Pascal's
triangle. Also, the pattern of counting the number of $\mathsf{Lie}$-brackets
of a particular Leibniz $n$-algebra leads us to a new property of Pascal's
triangle. Moreover, we discuss some results which improve the existing upper
bound published in [23] for $m$-dimensional $\mathsf{Lie}$-nilpotent Leibniz
$n$-algebras with $d$-dimensional $\mathsf{Lie}$-commutator. In particular, it
is shown that if $\mathfrak{q}$ is an $m$-dimensional $\mathsf{Lie}$-nilpotent
Leibniz $2$-algebra with one-dimensional $\mathsf{Lie}$-commutator, then
$\dim\mathcal{M}_{\mathsf{Lie}}(\mathfrak{q})\leq \frac{1}{2}m(m-1)-1.$
