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Stanley length of monomial complete intersection ideals

Prove that for every monomial complete intersection ideal I = (u1, …, um) ⊂ S = K[x1, x2, …, xn], generated by monomials with degrees d1 ≤ d2 ≤ … ≤ dm, the Stanley length satisfies slength(I) = 1 + d1 + d1d2 + ⋯ + d1d2⋯d_{m−1}.

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Background

The paper establishes an upper bound for the Stanley length of a monomial complete intersection I generated by monomials of degrees d1, …, dm: slength(I) ≤ 1 + d1 + d1d2 + ⋯ + d1d2⋯d_{m−1}. The authors conjecture that this upper bound is in fact exact for all such ideals, which would provide a precise formula for slength(I) in the complete intersection case and strengthen their general upper bound result.

References

We propose the following conjecture:

In the hypothesis of the previous theorem, we have that $$slength(I)= 1+d_1+d_1d_2+\cdots+d_1d_2\cdots d_{m-1}.$$

— On the Stanley length of monomial ideals (2507.17935 - Cimpoeas, 23 Jul 2025) in Conjecture, following Theorem 4 (Theorem \ref{t4}), Section “Main results”