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…

Proof Write $q_t=q(1,t)$, and let … The coefficient recurrences in the Zeroing Algorithm give … while for $0\leq t<k$ the initial values satisfy … Therefore the ordinary generating…

Proof Use zero-based indices $0\leq a,b<n$, and write $H=R_n(b_k(r))$. Then … Thus $H$ is the moment matrix of the weight $r(1-x)^{r-1}$ on $[0,1]$. For $k\geq0$, define the shifte…

Proof Let … where $r_j=1$ exactly when $j=2^a-1$ for some $a\geq0$, and $r_j=0$ otherwise. The shifted Rueppel sequence is an aeration of the original sequence: … Order the rows an…

Proof, valid for every $k\geq 3$ Let … be the $m$-th $k$-gonal-pyramidal number, and let $a_k(n)$ be the $n$-th positive integer not among these numbers. Set … I will prove the fol…

Counterexample at n=6 and likely pattern transposition The displayed equality is false as written. At $n=6$, exhaustive evaluation of the definitions gives … and … whereas … For th…

Counterexample to the fourth identity and source correction The fourth displayed identity is false as written. Take $t=n=1$, so $r=3$. Using … we obtain … and … Therefore the print…

Indexing correction and resolution The statement is false under its literal inclusive indexing. It defines $M_n$ using $0\le i,j\le n$, making $M_n$ an $(n+1)\times(n+1)$ matrix. A…

Degree-two counterexample The proposed product formula fails already for $m=2$ and $k=1$. The defining relations for the $(p,q)$-deformed Jordan plane include … Therefore … where…

Counterexample and off-by-one correction Write $q$ for the sequence index. Take … The conjecture as printed concerns the sequence $C_{q+2}$. Its first nontrivial Hankel determinant…

Counterexample to the formula as printed Take $n=1$ and $r=3$. Then … so $R_1(d_k)=[1]$, whose inverse is also $[1]$. For the proposed matrix $D(1,3)$, set $i=j=1$. The defining su…