Analysis of moments and cumulants in alternating sign matrices

In this work, we study the discrete observables $$E_k = \sum_{i,j=1}^n (i-j)^k A_{i,j}$$ associated with $n\times n$ alternating sign matrices $A = (A_{i,j})$. This work develops exact formulas for expectations using Bernoulli polynomials, exponential generating functions, expansions in $1/n$ linked to Riemann zeta functions, and cumulants up to fourth order via integrable kernel methods. All intermediate calculations, expansions, and pedagogical details are provided to illustrate the interplay between combinatorial sums, analytic expansions, and integrable structures in alternating sign matrices.