The classical Benjamin-Ono equation v_t + v v_x = e_1 J[v_{xx}] with periodic boundary conditions is a completely integrable system for v: T → R on the unit circle T, where J is the Hilbert transform. Let H denote the Hardy space on the circle, |0> in H the vacuum, and L(v) the Toeplitz operator with symbol v. Building on the work of Nazarov-Sklyanin (2013), we show that the infinitely-many \ell=0,1,2,3,… conservation laws T_{\ell}(v|e_1) of the classical Benjamin-Ono system are moments of a conserved density given by the spectral shift function of L(v) + e_1 d/dx and its (0,0) minor. Moreover, Nazarov-Sklyanin (2013) also treat the canonical quantization of this system, associating to classical conservation laws T_{\ell}(v|e_1) an infinite hierarchy of commuting operators \hat{T}_{\ell}(v|e_2,e_1) with \hbar = - e_1 e_2 simultaneously diagonalized on Jack polynomials with parameter \alpha = - e_1 / e_2. Their eigenvalues on Jacks depend only on the profile of the *anisotropic* Young diagram built from rectangles of size (e_2, e_1). Just like in the classical case, we realize the slopes of a profile of an anisotropic Young diagram as a spectral shift function in the sense of Krein (1953). The talk will begin with the general framework of Kerov’s Markov-Krein correspondence (1998). These constructions are new even for Schur polynomials e_1 + e_2 = 0 that relies on the first Szegö theorem (1915). As time permits, we introduce a family of combinatorial objects called ``ribbon paths” that are for random partitions what ribbon graphs are for random matrices at general \beta /2 = -(e_2/e_1) >0.