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.
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 de
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 a
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 Cox
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 o
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 lceilf