Let n be a non-negative integer and let k be a positive integer. A partition of n into no more than k parts is an expression of the form n = a1 + a2 + ... + ak, where a1 > = a2 > = ... > = ak > = 0. Let pk(n) denote the number of partitions of n into no more than k parts. Euler knew that the summation of pk(n)xn as n runs from 0 to infinity was equal to the formal power series expansion of the product of (1-xi)-1 as i runs from 1 to k. Alternatively, a partition of n into no more than k parts can also be viewed as an order-reversing map from a chain of k elements to the set of non-negative integers such that the sum of the images is equal to n. Let P be any fixed poset. In his thesis, R.P. Stanley extended this view by considering order reversing maps from P to the non-negative integers. If the images for such a map summed to n, he said that the map was a P-partition of n. He then associated the weight xn to each such map and summed over all such maps to form the P-partition generating function F(P,x) for the poset P. Stanley showed that F(P,x) for any poset P could always be written as a rational function of a certain form. The numerator polynomial was formed by compiling a certain combinatorial statistic from the many 'order extensions' of P. In rare cases, this numerator polynomial is a factor of the specified denominator polynomial, and the remaining denominator can be expressed as a product of factors of the form (1-x^hy) for certain integers hy. Stanley proved that this was possible for shapes in 1970. Sagan later introduced the term 'hook length poset' into the literature, since the integers hy in the case of shapes had turned out to be the 'hook lengths' defined by Frame, Robinson, and Thrall. There are many different (but closely related) reasonable possible definitions for hook length posets. One definition (made in the order dual context of reverse-P-partitions) appears in Sagan, European J. Combins 3 (1982). A slightly more demanding definition would require the formulation of a rule which would assign one integer hy to each element of P in such a way that the P-partition generating function would be equal to the product of the factors (1-x^hy)-1 as y runs through the elements of P. Our result with Dale Peterson for d-complete posets is actually much stronger, since it shows that a certain finer multivariate P-partition generating function factors nicely. Here the number of variables is equal to the number of elements in the top tree of P, which is a concept defined on the page describing the classification of d-complete posets.
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