A combinatorial interpretation of the integral of the product of Legendre polynomials

Denote by $P_n (x)$ the Legendre polynomial of degree n and let \[ I_{n_1 , \cdots ,n_k } = \int_{ - 1}^1 {P_{n_1 } (x)} \cdots P_{n_k } (x)\,dx. \]$I_{n_1 , \cdots ,n_k } $ is written as a sum involving binomial coefficients and the sum is interpreted via a combinatorial model. This makes possible a combinatorial proof of a number of general theorems concerning $I_{n_1 , \cdots ,n_k } $, not all of which seem analytically straightforward, including a direct combinatorial derivation of the known formula for $I_{a,b,c} $ and the expression of $I_{a,b,c,d} $ as a simple finite sum. In addition, a number of apparently new combinatorial identities are obtained.