Phagocytic monocyte-derived macrophages associate with the nodes of Ranvier and initiate demyelination while microglia clear debris and display a suppressed metabolic gene signature in EAE.

Extending the classical pop-stack sorting map on the lattice given by the right weak order on S n , Defant defined, for any lattice M , a map Pop M : M → M that sends an element x ∈ M to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study Pop M (M ) when M is the weak order of type B n , the Tamari lattice of type B n , the lattice of order ideals of the root poset of type A n , and the lattice of order ideals of the root poset of type B n . In particular, we settle four conjectures proposed by Defant and Williams on the generating functionwhere U M (b) is the set of elements of M that cover b.

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Bonichon and Morel first introduced d-permutations in their study of multidimensional permutations. Such permutations are represented by their diagrams on [n] d such that there exists exactly one point per hyperplane x i that satisfies x i = j for i ∈ [d] and j ∈ [n]. Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. Further, we relate 3-permutation avoidance classes with their respective recurrence relations, which lead to interesting combinatorial properties of these 3-permutation avoidance classes. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.

Weights of permutations were originally introduced by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation σ viewed as a sequence of integers, computing the weight of σ involves recursively counting descents of certain subpermutations of σ. Using this weight function, one can define a q-analog En(x, q) of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials En(x, q). First, we show that the coefficients of En(x, q) stabilize as n goes to infinity, which was conjectured by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24-49, 2019), and enables the definition of the formal power series W d (t), which has interesting combinatorial properties. Second, we derive a recurrence relation for En(x, q), similar to the known recurrence for the classical Eulerian polynomials An(x). Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.