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We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $Delta_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $Delta_{e_k} e_n|_{q=1}$ with any symmetric function.
The distribution of certain Mahonian statistic (called $mathrm{BAST}$) introduced by Babson and Steingr{i}msson over the set of permutations that avoid vincular pattern $1underline{32}$, is shown bijectively to match the distribution of major index o
The symmetric group $mathfrak{S}_n$ acts on the polynomial ring $mathbb{Q}[mathbf{x}_n] = mathbb{Q}[x_1, dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $mathfrak{S}_n$-invariant polynomials with vanishing
A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. v{C}erny conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most
We introduce a family of varieties $Y_{n,lambda,s}$, which we call the $Delta$-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,lambda,s})$ and show that there is a symm
The Markoff injectivity conjecture states that $wmapstomu(w)_{12}$ is injective on the set of Christoffel words where $mu:{mathtt{0},mathtt{1}}^*tomathrm{SL}_2(mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2