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Given a continuous ordinary Galois representation$\bar{\rho}:G_{\mathbb{Q}_p}\rightarrow\mathrm{GL}_n(\overline{\mathbb{F}}_p)$,Breuil and Herzig constructed an admissible smooth$\overline{\mathbb{F}}_p$-representation $\Pi(\bar{\rho})^{\mathrm{ord}}$ of$\mathrm{GL}_n(\mathbb{Q}_p)$ and showed that it occurs in certain globallydefined mod $p$ cohomology spaces. By applying Taylor-Wiles patching to spacesof ordinary automorphic representations we prove that the indecomposable piecesof $\Pi(\bar{\rho})^{\mathrm{ord}}$ each occur with the same multiplicity at awell-chosen tame level.

Multiplicities in the ordinary part of mod $p$ cohomology for $\mathrm{GL}_n(\mathbb{Q}_p)$