Partition Lattice q-Analogs Related to q-Stirling Numbers
Curtis Bennett
, Kathy J. Dempsey
and Bruce E. Sagan
DOI: 10.1023/A:1022459817380
Abstract
We construct a family of partially ordered sets (posets) that are q-analogs of the set partition lattice. They are different from the q-analogs proposed by Dowling [5]. One of the important features of these posets is that their Whitney numbers of the first and second kind are just the q-Stirling numbers of the first and second kind, respectively. One member of this family [4] can be constructed using an interpretation of Milne [9] for S[ n, k] as sequences of lines in a vector space over the Galois field F q. Another member is constructed so as to mirror the partial order in the subspace lattice.
Pages: 261–283
Keywords: set partition lattice; vector space over a finite field; $q$-Stirling number
Full Text: PDF
References
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2. Carlitz, L., "On Abelian fields," Trans, Amer. Math. Soc. 35 (1933), 122-136.
3. Carlitz, L., "q-Bernoulli numbers and polynomials," Duke Math. J. IS (1948), 987-1000.
4. Dempsey, K.J., "^-Analogs and Vector Spaces," Ph.D. thesis, Michigan State University, East Lansing, 1992.
5. Dowling, T.A., A q-analog of the partition lattice, in A Survey of Combinatorial Theory, J.N. Srivastava et al., eds., North-Holland Pub. Co., Amsterdam, 1973,101-115.
6. Gessel, I.M., "A g-analog of the exponential formula," Discrete Math. 40 (1982), 69-80.
7. Gould, H.W., "The ^-Stirling numbers of the first and second kinds," Duke Math. J. 28 (1961), 281-289.
8. Milne, S.C., "A g-analog of restricted growth functions, Dobinski's equality, and Charlier polynomials," Trans. Amer. Math. Soc. 245 (1978), 89-118.
9. Milne, S.C., "Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers," Adv. in Math. 43 (1982), 173-196.
10. Simion, R. "On q-analogues of partially ordered sets," to appear in J. Combin. Theory Ser. A.
11. Sagan, B.E., "A maj statistic for set partitions," European J. Combin. 12 (1991), 69-79.
12. Stanley, R.P., Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1986.
13. Wachs, M. and White, D. p, "^-Stirling numbers and set partition statistics," / Combin. Theory Ser. A 56 (1991), 27-46.