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Using the diagrammatic approach to integrals over Gaussian random matrices,we find a representation for polynomial Lie group integrals as infinite sumsover certain maps on surfaces. The maps involved satisfy a specific condition:they have some marked vertices, and no closed walks that avoid these vertices.We also formulate our results in terms of permutations, arriving at new kindsof factorization problems.

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations