Some combinatorial formulas for the partial r-Bell polynomials

Mark Shattuck
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 63—76
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Authors and affiliations

Mark Shattuck
Department of Mathematics, University of Tennessee
Knoxville, TN 37996, United States


The partial r-Bell polynomials generalize the classical partial Bell polynomials (coinciding with them when r = 0) by assigning a possibly different set of weights to the blocks containing the r smallest elements of a partition no two of which are allowed to belong to the same block. In this paper, we study the partial r-Bell polynomials from a combinatorial standpoint and derive several new formulas. We prove some general identities valid for arbitrary values of the parameters as well as establish formulas for some specific evaluations. Several of our results extend known formulas for the partial Bell polynomials and reduce to them when r = 0. Our arguments are largely combinatorial, and therefore provide, alternatively, bijective proofs of these formulas, many of which were shown by algebraic methods.


  • Partial Bell polynomial
  • r-Stirling number
  • Combinatorial identity

AMS Classification

  • 05A19
  • 05A18
  • 11B75


  1. Abbas, M. & Bouroubi, S. (2005) On new identities for Bell’s polynomials, Discrete Math., 293, 5–10.
  2. Belbachir, H., Bouroubi, S. & Khelladi, A. (2008) Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution, Ann. Math. Inform., 35, 21–30.
  3. Bell, E. T. (1934) Exponential polynomials, Ann. of Math., 35, 258–277.
  4. Birmajer, D., Gil, J. B. & Weiner, M. D. (2016) On the enumeration of restricted words over a finite alphabet, J. Integer Seq., 19, Art. 16.1.3.
  5. Broder, A. Z. (1984) The r-Stirling numbers, Discrete Math. 49, 241–259.
  6. Cheon, G.-S. & Jung, J.-H. (2012) r-Whitney numbers of Dowling lattices, Discrete Math. 312, 2337–2348.
  7. Comtet, L. (1970) Advanced Combinatorics, Presses Universitaires de France, Paris.
  8. Cvijovic, D. (2011) New identities for the partial Bell polynomials, Appl. Math. Lett. 24(9), 1544–1547.
  9. Eger, S. (2016) Identities for partial Bell polynomials derived from identities for weighted integer compositions, Aequationes Math. 90(2), 299–306.
  10. Mihoubi, M. & Rahmani, M. (2013) The partial r-Bell polynomials
  11. Nyul, G. & Racz, G. (2015) The r-Lah numbers, Discrete Math., 338, 1660–1666.
  12. Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences,
  13. Zhang, Z. & Yang, J. (2012) Notes on some identities related to the partial Bell polynomials, Tamsui Oxf. J. Inf. Math. Sci. 28(1), 39–48.

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Cite this paper

Shattuck, M. (2017). Some combinatorial formulas for the partial r-Bell polynomials. Notes on Number Theory and Discrete Mathematics, 23(1), 63-76.

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