We investigate a generalization of stacks that we call C-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that C-machines generate, and how these systems of functional equations can be iterated and sometimes solved. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by C-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their counting sequences, seem to not have D-finite generating functions.

Is the susceptibility of the Ising model differentially algebraic?2 Key-words:non-holonomic functions, differentially algebraic functions, differentially transcendental functions, closure properties, non-linear differential equations, susceptibility of the Ising model, modulo prime calculations, algebraic functions, composition of functions, diagonals of rational functions, algebraic power series. ‡ No rigorous proof of this result exists, but no reasonable person doubts it. § These are known to be D-finite -see Appendix A. † D-finite functions can be expressed as solutions of (an infinite number of) differentially algebraic equations with movable singularities. However, that D-finite functions can also be solutions of differentially algebraic equations with fixed critical points is quite remarkable.

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We prove that the set of patterns {1324, 3416725} is Wilf-equivalent to the pattern 1234 and that the set of patterns {2143, 3142, 246135} is Wilf-equivalent to the set of patterns {2413, 3142}. These are the first known unbalanced Wilf-equivalences for classical patterns between finite sets of patterns.

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length n patterns in the separable permutations is equal to the number of partitions of n − 1.These operations are more naturally understood graphically: the graph of σ ⊕ τ (resp., σ ⊖ τ) is obtained by stacking the graph of τ above (resp., below) and to the right of that of σ. Both operations are individually, but not jointly, associative and are noncommutative. This paper is concerned with separable permutations which are defined as the class of all permutations that can be formed from the length one permutation, 1, by iterated applications of direct and skew sums. For instance, π = 543612 is separable, as:A permutation class is a set of permutations that forms a downset in the pattern ordering, i.e., a set C for which π ∈ C and σ ≺ π implies σ ∈ C. The class of all permutations is denoted S. Enumerating