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Lozenge Tilings of a Hexagon with a Horizontal Intrusion

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Abstract

Motivated by a conjecture posed by Fulmek and Krattenthaler, we provide product formulas for the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion. As a direct corollary, we obtain a product formula for the number of boxed plane partitions with a certain restriction. We also investigate the asymptotic behavior of the ratio between the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion and that of a semiregular hexagon without an intrusion.

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Notes

  1. A lozenge is a union of two adjacent unit triangles. A lozenge tiling of a region is a collection of lozenges that cover the region without any gaps or overlaps. For example, see the left picture in Fig. 3 or the left picture in Fig. 5.

  2. For a positive integer n,  \(n!!:= {\left\{ \begin{array}{ll} n\cdot (n-2)\cdots 4\cdot 2 &{} \text {if } n \text { is even},\\ n\cdot (n-2)\cdots 3\cdot 1 &{} \text {if } n \text { is odd}. \end{array}\right. }\)

  3. This extension is based on the fact that the shifted factorial \((a)_n\) and the quotient of two gamma functions \(\frac{\Gamma (a+n)}{\Gamma (a)}\) agree for a positive integer (or half-integer) a and a nonnegative integer n. One can check that the definition of \((a)_{n}\) for a negative integer n (=\(\frac{1}{(a-1)(a-2)\cdots (a+n)}\)) also coincides with the value of \(\frac{\Gamma (a+n)}{\Gamma (a)}.\)

  4. It is \( M(H_{a,b,c})=\prod _{i=1}^{a}\prod _{j=1}^{b}\prod _{k=1}^{c}\frac{i+j+k-1}{i+j+k-2}=\frac{H(a)H(b)H(c)H(a+b+c)}{H(a+b)H(b+c)H(c+a)},\) where \(H(n):= \prod _{i=0}^{n-1} i!.\)

  5. \(n! \sim \sqrt{2\pi n}(\frac{n}{e})^n\) as \(n\rightarrow \infty .\)

  6. Throughout this paper, \(a(N)\sim b(N)\) (as \(N\rightarrow \infty \)) means \( \lim _{N\rightarrow \infty }\frac{a(N)}{b(N)}=1.\)

  7. For example, if one take natural logarithm to the latter inequality \(K_{2}(a,c,c,d)< 1,\) then one can see that it is enough to show that \(g(a+d)>g(a)+g(d)\) and \(g(a+c)+g(c-d)>g(a+c-d)+g(c)\) hold. These two inequalities can be verified using Jensen’s inequality and the fact that \(\lim _{x\rightarrow 0^{+}}g(x)=0.\)

  8. A perfect matching is a subset of edges of a bipartite graph such that every vertex is incident to precisely one edge. See the right picture in Fig. 9 for an example.

  9. We can do this division because the left side of (4.13) is nonzero because of the case when \(c-b=0.\)

  10. We can do this division because the left side of (4.11) is nonzero because of the case when \(c-b=0.\)

  11. \(G(n+1) \sim A^{-1}(2\pi )^{\frac{n}{2}}n^{\frac{1}{2}n^2-\frac{1}{12}}e^{-\frac{3}{4}n^{2}+\frac{1}{12}}\) as \(n\rightarrow \infty .\) Here, A is the Glaisher–Kinkelin constant.

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Acknowledgements

The author thanks his advisor Professor Mihai Ciucu for his continuing encouragement and motivation. The author also thanks anonymous reviewers for carefully reading the original version of the paper and giving helpful comments. David Wilson’s program vaxmacs was helpful when the author tried to find the formulas.

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  1. Department of Mathematics, Indiana University, Bloomington, USA

    Seok Hyun Byun

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Byun, S.H. Lozenge Tilings of a Hexagon with a Horizontal Intrusion. Ann. Comb. 26, 943–970 (2022). https://doi.org/10.1007/s00026-022-00595-2

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