Translated Whitney and r-Whitney Numbers: A Combinatorial Approach
نویسندگان
چکیده
Using a combinatorial approach, we introduce the translated Whitney numbers. This seems to be more natural than to write a product of anarithmetical progression in terms of a power variable and conversely. We also extend our ideas to translated r-Whitney numbers of both kinds and to translated Whitney-Lah numbers.
منابع مشابه
Eulerian Numbers Associated with Arithmetical Progressions
In this paper, we give a combinatorial interpretation of the r-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to r-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruen...
متن کاملSome Theorems and Applications of the (q, r)-Whitney Numbers
The (q, r)-Whitney numbers were recently defined in terms of the q-Boson operators, and several combinatorial properties which appear to be q-analogues of similar properties were studied. In this paper, we obtain elementary and complete symmetric polynomial forms for the (q, r)-Whitney numbers, and give combinatorial interpretations in the context of A-tableaux. We also obtain convolution-type ...
متن کاملThe Translated Dowling Polynomials and Numbers
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated D...
متن کاملA generalized recurrence formula for Stirling numbers and related sequences
In this note, we provide a combinatorial proof of a generalized recurrence formula satisfied by the Stirling numbers of the second kind. We obtain two extensions of this formula, one in terms of r-Whitney numbers and another in terms of q-Stirling numbers of Carlitz. Modifying our proof yields analogous formulas satisfied by the r-Stirling numbers of the first kind and by the r-Lah numbers.
متن کاملOn q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers
We define the (q, r)-Whitney numbers of the first and second kinds in terms of the q-Boson operators, and obtain several fundamental properties such as recurrence formulas, orthogonality and inverse relations, and other interesting identities. As a special case, we obtain a q-analogue of the r-Stirling numbers of the first and second kinds. Finally, we define the (q, r)-Dowling polynomials in t...
متن کامل