A partition u of [k] = f1; 2; : : : ; kg is contained in another partition v of [l] if [l] has a k-subset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of Stanley{Wilf conjecture is proposed. We prove that the GF counting v is rational ...

Let S(n, k) be the classical Stirling numbers of the second kind, d > 1 be an integer, and P,Q,R ∈ Q[X1, . . . , Xm] be nonconstant polynomials such that P does not divide Q and R is not a dth power. We prove that if k1, . . . , km are any sufficiently large distinct positive integers then, setting Si = S(n, ki), Q(S1,...,Sm) P (S1,...,Sm) ∈ Z for only finitely many n ∈ N and R(S1, . . . , Sm) ...

We prove that for any k = 1, . . . , 2 the 2-adic order of the Stirling number S(2, k) of the second kind is exactly d(k) − 1, where d(k) denotes the number of 1’s among the binary digits of k. This confirms a conjecture of Lengyel.

The normal ordering of an integral power the number operator in terms boson operators is expressed with help Stirling numbers second kind. As a ‘degenerate version’ this, we consider degenerate operators, which represented by means application this ordering, derive two equations defining kind and Dobinski-like formula for Bell polynomials.

Many works have been done in recent years as to explorations for degenerate versions of some special polynomials and numbers, which began with the pioneering work Carlitz on Bernoulli Euler polynomials. This paper is focused study properties, recurrence relations identities related r-Bell polynomials, two variable Fubini r-Stirling numbers second kind. Especially, we express power series ∑n=0∞∑...