## Rook theory, generalized Stirling numbers and (p,q)-analogues (2004)

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Citations: | 8 - 1 self |

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@MISC{Remmel04rooktheory,,

author = {J. B. Remmel and Michelle L. Wachs},

title = { Rook theory, generalized Stirling numbers and (p,q)-analogues },

year = {2004}

}

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### Abstract

In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S 1 (α, β, r) andS 2 (α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β =0andα and r are nonnegative integers both of our (p, q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our (p, q)-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our (p, q)-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

### Citations

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116 |
The Theory of Partitions, Encyclopedia of Mathematics and its
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- 1976
(Show Context)
Citation Context ...usinthecasewhereγ = n is a non-negative integer, is the usual (p, q)-analogue of n. Wealsolet [γ]p,q = pγ − qγ . (16) p − q [n]p,q = q n−1 + pq n−2 + ···+ p n−2 q + p n−1 [n]p,q! =[n]p,q[n − 1]p,q ···=-=[1]-=-p,q and � � n [n]p,q! = k p,q [k]p,q![n − k]p,q! . � � � � n n We shall write [n]q, [n]q! and for [n]1,q, [n]1,q! and respectively. k k q 1,q For the type I (p, q)-analogues of (8) and (9), we replace... |

86 | Théorie géométrique des polynômes eulériens - Foata, Schützenberger - 1970 |

57 | Weyl groups, the hard Lefschetz theorem, and the Sperner property
- Stanley
- 1980
(Show Context)
Citation Context ...h is just the usual inversion statistic 1≤s<t≤n χ(σ(s) >σ(t)). This is precisely what inv reduces to when (i, j) =(0, 1). For the hyperoctahedral group the Coxeter length is described as follows (cf. =-=[24]-=-): l(σ, e) = � χ(σ(s) >σ(t) & et =0) + + 1≤s<t≤n � 1≤s<t≤n n� ett t=1 χ(σ(s) <σ(t) & et =1) Clearly, our inv statistic does not reduce to length when (i, j) =(1, 2). However, we can modify our definit... |

46 |
q-Stirling numbers and set partition statistics
- WACHS, WHITE
- 1990
(Show Context)
Citation Context ...ions in terms of rook placements and restricted growth functions. A more general two parameter, (p, q)-analogue of the Stirling number of the second kind was introduced and studied by Wachs and White =-=[26]-=-, who also gave interpretations in terms of rook placements and restricted growth functions. We shall define two natural (p, q)-analogues of the S i n,k(α, β, r)’s, one of which reduces to the (p, q)-... |

38 |
q-Counting rook configurations and a formula of Frobenius
- Garsia, Remmel
- 1986
(Show Context)
Citation Context ...n, S 1 n,k(α, β, r) =S 2 n,k(β,α,−r). (5) q-Analogues of the Stirling numbers of the first and second kind were first considered by Gould [14] and further studied by Milne [21][20], Garsia and Remmel =-=[11]-=-, and others, who gave interpretations in terms of rook placements and restricted growth functions. A more general two parameter, (p, q)-analogue of the Stirling number of the second kind was introduc... |

37 |
A q-analog of restricted growth functions, Dobinski's equality, and Chartier polynomials
- Milne
- 1978
(Show Context)
Citation Context ...and (2)thatforall0≤ k ≤ n, S 1 n,k(α, β, r) =S 2 n,k(β,α,−r). (5) q-Analogues of the Stirling numbers of the first and second kind were first considered by Gould [14] and further studied by Milne [21]=-=[20]-=-, Garsia and Remmel [11], and others, who gave interpretations in terms of rook placements and restricted growth functions. A more general two parameter, (p, q)-analogue of the Stirling number of the ... |

30 |
Restricted growth functions, rank row matchings of partition lattices, and q-Stirling
- Milne
- 1982
(Show Context)
Citation Context ...(1) and (2)thatforall0≤ k ≤ n, S 1 n,k(α, β, r) =S 2 n,k(β,α,−r). (5) q-Analogues of the Stirling numbers of the first and second kind were first considered by Gould [14] and further studied by Milne =-=[21]-=-[20], Garsia and Remmel [11], and others, who gave interpretations in terms of rook placements and restricted growth functions. A more general two parameter, (p, q)-analogue of the Stirling number of ... |

27 | The problem of the rooks and its applications - Kaplansky, Riordan - 1946 |

25 | D.E.White Rook theory I: Rook equivalence of Ferrers boards - Goldman, Joichi - 1975 |

20 |
A unified approach to generalized Stirling numbers
- Hsu, Shiue
- 1998
(Show Context)
Citation Context ...A19,05A30 Abstract In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S 1 (α, β, r) andS 2 (α, β, r) as introduced by Hsu and Shiue =-=[17]-=-. We show that in the case where β =0andα and r are nonnegative integers both of our (p, q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions f... |

19 |
Interpolating Set Partition Statistics
- White
- 1994
(Show Context)
Citation Context ...t in the case when i =0andj =1,ourtypeI(p, q)-Stirling numbers of the first and second kind, s 0,1 n,k (p, q) andS0,1 n,k (p, q), have been studied by a number of other authors, see [18], [19], [27], =-=[28]-=- and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], [13]. Moreover an alternative approach to combinatorially interpreting a di... |

17 | q-rook polynomials and matrices over finite fields - Haglund - 1998 |

17 |
Log concave sequences of symmetric functions and analogs of the JacobiTrudi determinants
- Sagan
- 1992
(Show Context)
Citation Context ...case when i =0andj =1,ourtypeI(p, q)-Stirling numbers of the first and second kind, s 0,1 n,k (p, q) andS0,1 n,k (p, q), have been studied by a number of other authors, see [18], [19], [27], [28] and =-=[23]-=-. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], [13]. Moreover an alternative approach to combinatorially interpreting a different f... |

16 |
The q-Stirling numbers of first and second
- Gould
- 1961
(Show Context)
Citation Context ... it is easy to see from equations (1) and (2)thatforall0≤ k ≤ n, S 1 n,k(α, β, r) =S 2 n,k(β,α,−r). (5) q-Analogues of the Stirling numbers of the first and second kind were first considered by Gould =-=[14]-=- and further studied by Milne [21][20], Garsia and Remmel [11], and others, who gave interpretations in terms of rook placements and restricted growth functions. A more general two parameter, (p, q)-a... |

15 |
On Whitney numbers of Dowling lattices
- Benoumhani
- 1996
(Show Context)
Citation Context ...,k (p, q), have been studied by a number of other authors, see [18], [19], [27], [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see =-=[2]-=-, [3], [13]. Moreover an alternative approach to combinatorially interpreting a different family of generalized (p, q)-Stirling numbers which includes our (p, q)-Stirling numbers s i,j n,k (p, q) andS... |

13 | An interpretation for Garsia and Remmel’s q-hit numbers - Dworkin - 1996 |

12 |
On some numbers related to Whitney numbers of Dowling lattices
- Benoumhani
- 1997
(Show Context)
Citation Context ..., q), have been studied by a number of other authors, see [18], [19], [27], [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], =-=[3]-=-, [13]. Moreover an alternative approach to combinatorially interpreting a different family of generalized (p, q)-Stirling numbers which includes our (p, q)-Stirling numbers s i,j n,k (p, q) andSi,j n... |

8 |
m-rook numbers and a generalization of a formula of Frobenius to Cm
- Briggs, Remmel
- 2006
(Show Context)
Citation Context ...natorial theory of hit polynomials in that model. Finally, certain special cases of the more general (p, q) rooknumbers˜r j k,B (1,q) show up in yet another rook theory model due to Briggs and Remmel =-=[4, 7]-=- where the rook placements naturally correspond to elements of the wreath product of the cyclic group Zk and the symmetric group Sn, Zk§Sn. Again there is a natural combinatorial theory of hit polynom... |

8 | Generalized Stirling Numbers, Convolution Formulae and p,q-Analogues
- Medicis, Leroux
- 1995
(Show Context)
Citation Context ...lso note that in the case when i =0andj =1,ourtypeI(p, q)-Stirling numbers of the first and second kind, s 0,1 n,k (p, q) andS0,1 n,k (p, q), have been studied by a number of other authors, see [18], =-=[19]-=-, [27], [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], [13]. Moreover an alternative approach to combinatorially inter... |

6 |
A unified combinatorial approach for q-(and p, q-)Stirling numbers
- Médicis, Leroux
- 1993
(Show Context)
Citation Context ...ould also note that in the case when i =0andj =1,ourtypeI(p, q)-Stirling numbers of the first and second kind, s 0,1 n,k (p, q) andS0,1 n,k (p, q), have been studied by a number of other authors, see =-=[18]-=-, [19], [27], [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], [13]. Moreover an alternative approach to combinatorially... |

6 |
σ-Resticted growth functions and p, q-Stirling numbers
- Wachs
- 1994
(Show Context)
Citation Context ...te that in the case when i =0andj =1,ourtypeI(p, q)-Stirling numbers of the first and second kind, s 0,1 n,k (p, q) andS0,1 n,k (p, q), have been studied by a number of other authors, see [18], [19], =-=[27]-=-, [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], [13]. Moreover an alternative approach to combinatorially interpretin... |

5 |
A Combinatorial Interpretation for p, q-Hit Numbers, preprint
- BRIGGS
(Show Context)
Citation Context ...iggs and Remmel [6] showed that when B = B(a1,...,an) is a Ferrers board, i.e. 0 ≤ a1 ≤ ... ≤ an ≤ n, thenhk,n(B,p,q) is polynomial in p and q with non-negative integer coefficients. Moreover, Briggs =-=[4, 5]-=- has shown that if Hk,n(B) is the set of all placements P in N 1 N (Bn) such that P has exactly k rooks in B, then there are statistics αB(P) andβB(P) such that hk,n(B,p,q) = � p αB(P) βB(P) q . P∈Hk,... |

5 | Rook theory for perfect matchings
- Haglund, Remmel
- 2001
(Show Context)
Citation Context ...(σ)+1 �n i=0 (1 − xqipn−i ) i∈Des(σ) comaj(σ) = � i∈Rise(σ) i. (146) Certain special cases of the rook numbers ˜r 2 k,B (1,q) also have shown up in another rook theory model due to Haglund and Remmel =-=[16]-=- where the rook placements naturally correspond the electronic journal of combinatorics 11 (2004), #R84 46sto partial perfect matchings in the complete graph K2n. Haglund and Remmel also develop a com... |

4 |
Q-Analogues and P, Q-Analogues of Rook Numbers and Hit Numbers and Their Extensions
- Briggs
- 2003
(Show Context)
Citation Context ...iggs and Remmel [6] showed that when B = B(a1,...,an) is a Ferrers board, i.e. 0 ≤ a1 ≤ ... ≤ an ≤ n, thenhk,n(B,p,q) is polynomial in p and q with non-negative integer coefficients. Moreover, Briggs =-=[4, 5]-=- has shown that if Hk,n(B) is the set of all placements P in N 1 N (Bn) such that P has exactly k rooks in B, then there are statistics αB(P) andβB(P) such that hk,n(B,p,q) = � p αB(P) βB(P) q . P∈Hk,... |

4 |
Cohomology of Dowling lattices and
- Gottlieb, Wachs
(Show Context)
Citation Context ... have been studied by a number of other authors, see [18], [19], [27], [28] and [23]. The case i = p = q = 1 has also appeared in the literature as Whitney numbers for Dowling lattices, see [2], [3], =-=[13]-=-. Moreover an alternative approach to combinatorially interpreting a different family of generalized (p, q)-Stirling numbers which includes our (p, q)-Stirling numbers s i,j n,k (p, q) andSi,j n,k (p,... |

2 |
A rook theory model for the p; q-analogues of Hsu and Shuiue's Generalized Stirling Numbers, in preparation
- Briggs, Remmel
(Show Context)
Citation Context ... (34) from (35) by using the fact that the the matrices ||s i,j n,k (p, q)|| and ||Si,j n,k (p, q)|| are inverses of each other. A direct combinatorial proof of (34) was found by Briggs and Remmel in =-=[8]-=-. We give a direct combinatorial proof the matrices ||s i,j n,k (p, q)|| and ||Si,j n,k (p, q)|| are inverses of each other. That is, if we start with our combinatorial interpretations of c i,j n,k (p... |

1 |
A(p, q)-analogue of a formula of Frobenius
- Briggs, Remmel
(Show Context)
Citation Context ... . Our model allows rooks in a given board to cancel cells not only on its own board but also on its companion board. We should also note that in the special case when i =0andj = 1, Briggs and Remmel =-=[6]-=- showed that there is a (p, q)-analogue of the hit polynomial corresponding to the rook number ˜r 1 n−k,B (p, q). That is, given a board B contained in the n × n board Bn, we define the p, q-hit polyn... |