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Using Jack polynomials, Goulden and Jackson have introduced a one parameterdeformation $\tau_b$ of the generating series of bipartite maps, whichgeneralizes the partition function of $\beta$-ensembles of random matrices. TheMatching-Jack conjecture suggests that the coefficients $c^\lambda_{\mu,\nu}$of the function $\tau_b$ in the power-sum basis are non-negative integerpolynomials in the deformation parameter $b$. Do{\l}\k{e}ga and F\'eray haveproved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namelythat coefficients $c^\lambda_{\mu,\nu}$ are in $\mathbb{Q}[b]$. In this paper,we prove the "integrality" part, i.e that the coefficients$c^\lambda_{\mu,\nu}$ are in $\mathbb{Z}[b]$.  The proof is based on a recent work of the author that deduces theMatching-Jack conjecture for marginal sums from an analog result for the$b$-conjecture, established in 2020 by Chapuy and Do{\l}\k{e}ga. A key step inthe proof involves a new connection with the graded Farahat-Higman algebra.

Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra