Martin Tassy - Dartmouth College The celebrated hook-length formula of Frame, Robinson, and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for counting tableaux of skew shapes. In 2014, Naruse announced a formula for this number that can be interpreted as a weighted partition function over lozenge tilings. Simulations by Morales, Pak and Panova in 2017 revealed that these weighted lozenge tilings exhibit limit shapes. We explain how this particular limiting behavior can be interpreted as the consequence of a variational principle. We will also explain how this variational principle gives us the existence and an interpretation of the first term in the asymptotics of the number of tableaux of a family of skew shapes, settling a conjecture of Morales, Pak, and Panova. Finally, we will attempt to characterize which types of combinatorial objects must satisfy such variational principles.