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We study the category $\mathcal{F}_n$ of finite-dimensional integrablerepresentations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. Wedefine an action of the Temperley--Lieb algebra with infinitely many generatorsand defining parameter $0$ on the category $\mathcal{F}_n$ by translationfunctors. We also introduce combinatorial tools, called weight diagrams andarrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$.Using the Temperley--Lieb algebra action and the combinatorics of weight andarrow diagrams, we then calculate the multiplicities of standard and costandardmodules in indecomposable projective modules and classify the blocks of$\mathcal{F}_n$. We also prove that indecomposable projective modules in thiscategory are multiplicity-free.

Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$