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Register a new userA determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata
محدد لأرقام دورة سترلينج تحتسب الآليات الآوتوماتية الوحيدة المصدر الغير مسمى الغير الدورية
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Authors
David Callan
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نظرنا إلى أن المحدد لأرقام دورة سترلينج تحتسب الآليات المصادقة المصدر واحد بدون تسمية غير حلقية. الإثبات يتضمن تعادلا من هذه الآليات إلى بعض المسارات الشبكية المشار إليها وتحويلا معاكسا للتأكيد لتقدير المحدد.
We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant.
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We give combinatorial proofs of $q$-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitzs identity, a new proof of the $q$-Frobenius identity of Garsia and Remmel and of Ehrenborgs Hankel $q$-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan-Lusztig polynomials.
The distribution of certain Mahonian statistic (called $mathrm{BAST}$) introduced by Babson and Steingr{i}msson over the set of permutations that avoid vincular pattern $1underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturally. The latter of which is shown to be related to the recently proven Delta Conjecture. During the course, a refined relation between $mathrm{BAST}$ and its reverse complement $mathrm{STAT}$ is derived as well.
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $frac{8}{9}n+112le mle lceilfrac{3n}{2}rceil+1$ and $n geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {it Ars Combin.,} {bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k geq 3$. Namely, we prove that for each $k geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {it J.~London Math.~Soc.,} {bf 18} (1978), 392--396).
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