Samuel Schlesinger

The statement as written is false already at $t=1$. For $t=1$, the poset $EN_{3,1}$ is a three-element chain. Its unique linear extension is $321$, which has three inversions and a…

This conjecture was proved by Colin Defant in Theorem 3.2 of “Proofs of Conjectures about Pattern-Avoiding Linear Extensions”: https://arxiv.org/abs/1905.02309 Published version: h…

This conjecture was proved by Colin Defant in Theorem 3.3 of “Proofs of Conjectures about Pattern-Avoiding Linear Extensions”: https://arxiv.org/abs/1905.02309 Published version: h…

This conjecture has been proved in later literature. The identities in this MathDB entry are exactly Conjecture 10.17 of Eric S. Egge: https://arxiv.org/abs/2208.01484 Yujie Du and…

Proof The MathDB status reflects the original 2023 version of the source. The current version proves this identity as Theorem 3.8. Here is also a direct formal-power-series proof.…

Proof Here the primes denote complementary modulus, not differentiation: $K'(x)=K(\sqrt{1-x^2})$ and $E'(x)=E(\sqrt{1-x^2})$. First consider the complementary moment. Make the subs…

Proof with an encoding-explicit binary reduction The conjecture is proved as Theorem 1 of Yongchun Li's Strong Formulations and Algorithms for Regularized A-optimal Design. Since r…

Proof (published by Shitov in 2013) This conjecture was resolved by Yaroslav Shitov in On the Complexity of Boolean Matrix Ranks. Here is the reduction. Let $G=(V,E_G)$ be a simple…

Counterexample to the literal arbitrary-family statement Let $\Psi$ be the family of graphs containing no cycles of length $4$ or $5$, where the forbidden cycles need not be induce…

Counterexample The identity is false already at $m=1$. Take … These are valid nonsingular values for the source variables, since … and $b_2,z,w$ are all nonzero. First evaluate the…