#### Document Type

Article

#### Publication Date

6-2015

#### Abstract

Let *S* = (*s*1, *s*2, . . .) be any sequence of nonnegative integers and let *S _{k}* = ∑

*=1*

^{k}_{i}*s*. We then define the falling (rising) factorials relative to

_{i}*S*by setting (

*x*) ↓

_{k,S}=(

*x*-

*S*

_{1})(

*x*-

*S*

_{2}) · · · (

*x*-

*S*) and (

_{k}*x*) ↑

*= (*

_{k,S}*x*+

*S*

_{1})(

_{x}+

*S*

_{2}) · · · (

*x*+

*S*) if

_{k}*k*≥ 1 with (

*x*) ↓

_{0,S}= (

*x*) ↑

_{0,S}= 1. It follows that {(

*x*) ↓

_{k,S}}

_{k>0}and {(

*x*)↑

_{k,S}}

_{k>0}are bases for the polynomial ring

**Q**[

*x*]. We use a rook theory model due to Miceli and Remmel to give combinatorial interpretations for the connection coefficients between any two of the bases {(

*x*)↓

_{k,S}}

_{k≥0}, {(

*x*)↑

_{k,S}}

_{k≥0}, {(

*x*)↓

_{k,Τ}}

_{k≥0}and {(

*x*)↑

_{k,Τ}}

_{k≥0}for any two sequences of nonnegative integers

*S*= (

*s*

_{1},

*s*

_{2}, . . .) and

*Τ*= (

*t*

_{1},

*t*

_{2}, . . .). We also give two different

*q*-analogues of such coefficients. Moreover, we use this rook model to give an alternative combinatorial interpretation of such coefficients in terms of certain pairs of colored permutations and set partitions with restricted insertion patterns.

#### Document Object Identifier (DOI)

10.1007/s00026-015-0268-7

#### Publisher

Springer Basel

#### Repository Citation

Liese, J., Miceli, B.K., & Remmel, J. (2015). Connection coefficients between generalized rising and falling factorial bases. *Annals of Combinatorics, 19*(2), 337-361. doi:10.1007/s00026-015-0268-7

#### Publication Information

Annals of Combinatorics