G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....it is negative must be negated to obtain the desired (necessarily non negative) answer. It is all a bit contrived, and reminiscent of the situation of matrix theory before vector spaces and linear transformations assumed their proper, foundational role. Greg Kuperberg s way of looking at things [K6] (reducing combinatorics to algebra) seems to lift up the corner of the tangled carpeting, to show the smooth oor beneath; I hope to supplement his e orts by doing the reverse, and nding a purely combinatorial way to understand the algebra of Kasteleyn matrices and their inverses. The rudiments ....

G. Kuperberg, An exploration of the permanent-determinant method, preprint.

....no subtraction; so, if all weights are positive at the start, edge weights or cell weights equal to 0 cannot arise later, and more generally, correction terms involving higher powers of e can never be canceled out additively. 2. These algorithms arose partly in response to work of Kuperberg (see [13]) which took a more algebraic perspective on enumeration, using the approach pioneered by Kasteleyn [10] Kasteleyn s method requires making some arbitrary choices that end up not affecting the final answers to meaningful enumerative questions. This new work arose out of an attempt to find the ....

Greg Kuperberg, An exploration of the permanentdeterminant method, math.CO/9810091.

....formula hunting, and the debatable issue of whether the occurrence of a single larger than expected prime factor rules out the existence of a product formula. For an example of a number whose roundness lies in this gray area, see the table of numbers given in Problem 8. It is worth noting that Kuperberg [1998, Section VII A] has shown that rigorous proofs of roundness need not always yield explicit product formulas. Christian Krattenthaler has written a Mathematica program called RATE that greatly expedites the process of guessing patterns in experimental data on enumeration of matchings; see ....

....of a bipartite planar graph as the determinant of a signed version of the bipartite adjacency matrix. In the case of lozenge tilings of hexagons and the associated matchings, it turns out that there is no need to modify signs of entries; the ordinary bipartite adjacency matrix will do. Greg Kuperberg [1998] has noticed that when row reduction and column reduction are systematically applied to the Kasteleyn Percus matrix of an a, b, c semiregular hexagon, one can obtain the b by b Carlitz matrix [Carlitz and Stanley 1975] whose (i, j) th entry is a c a i j . This matrix can also be recognized ....

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G. Kuperberg, "An exploration of the permanent-determinant method", Electron. J. Combin. 5 (1998), R46.

....lower border of the region is Gamma1, the weight of all other lozenges being 1. In Figure 6 the black lozenge has weight Gamma1, all other lozenges have weight 1. Yet another way to obtain this weight is through the perfect matchings point of view of lozenge tilings, elaborated for example in [21, 22]. In this setup, the cyclically symmetric lozenge tilings that we consider here 12 M. CIUCU, T. EISENK OLBL, C. KRATTENTHALER AND D. ZARE correspond bijectively to perfect matchings in a certain hexagonal graph (basically, the dual graph of a fundamental region of the cored hexagon) ....

G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.

.... (or appear to have) beautiful solutions (see e.g. 5, 6, 7, 8, 9, 13, 24, 42, 44] Second, these problems are very often related to the theory of symmetric functions and or the representation theory of classical and quantum Lie algebras, and to statistical physics (sometimes in disguise; see e.g. [10, 15, 17, 25, 26, 34, 38, 39, 40]) As is well known, the total number of rhombus tilings of a hexagon with side lengths a; b; c; a; b; c equals a Y i=1 b Y j=1 c Y k=1 i j k Gamma 1 i j k Gamma 2 : 1.1) This follows from MacMahon s enumeration [28, Sec. 429, q 1; proof in Sec. 494] of all plane ....

G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.

....lozenges being 1. In Figure 6 the black lozenge has 12 M. CIUCU, T. EISENK OLBL, C. KRATTENTHALER AND D. ZARE weight Gamma1, all other lozenges have weight 1. Yet another way to obtain this weight is through the perfect matchings point of view of lozenge tilings, elaborated for example in [21, 22]. In this setup, the cyclically symmetric lozenge tilings that we consider here correspond bijectively to perfect matchings in a certain hexagonal graph (basically, the dual graph of a fundamental region of the cored hexagon) Assignment of weights to the edges of this graph so that each face has ....

G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.

....prime factor rules out the existence of a product formula. For an example of a number whose roundness lies in this gray area, see the table of numbers given in problem 8. It is amusing to note that one can have rigorous proofs of roundness that do not yield explicit product formulas; see [Ku2], section VII A for some examples. A great source of the appeal of research on enumeration of matchings is the ease with which undergraduate research assistants can participate in the hunt for formulas and proofs; many members of the M.I.T. Tilings Research Group (composed mostly of undergraduates ....

....graph as the determinant of a signed version of the bipartite adjacency matrix. In the case of lozenge tilings of hexagons and the associated matchings, it turns out that there is no need to modify signs of entries; the ordinary bipartite adjacency matrix will do. Greg Kuperberg has noticed (see [Ku2]) that when row reduction and column reduction are systematically applied to the Kasteleyn Percus matrix of an a; b; c semiregular hexagon, one can obtain the b by b Carlitz matrix (see [CS] whose i; jth entry is i a c a i Gammaj j . This matrix can also be recognized as the Gessel Viennot ....

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G. Kuperberg, An exploration of the permanent-determinant method, preprint; math.CO/9810091.

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G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.

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G. Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), #R46, 34 pp.

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