On a conjecture about enumerating (2+2)(2+2)-free posets

Recently, Kitaev and Remmel posed a conjecture concerning the generating function for the number of unlabeled $(2+2)$-free posets with respect to number of elements and number of minimal elements. In this paper, we present a combinatorial proof of this conjecture

On a refinement of Wilf-equivalence for permutations

Recently, Dokos et al. conjectured that for all $k, m\geq 1$, the patterns $12\ldots k(k+m+1)\ldots (k+2)(k+1)$ and $(m+1)(m+2)\ldots (k+m+1)m\ldots 21$ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\geq 1$ and $m=1$. In fact, we construct a descent set preserving bijection between $12\ldots k (k-1)$-avoiding permutations and $23\ldots k1$-avoiding permutations for all $k\geq 3$. As a corollary, our bijection enables us to settle a conjecture of Gowravaram and Jagadeesan concerning the Wilf-equivalence for permutations with given descent sets