Skip to main content
Log in

Identities for partial Bell polynomials derived from identities for weighted integer compositions

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript


We discuss closed-form formulas for the (n, k)th partial Bell polynomials derived in Cvijović (Appl Math Lett 24:1544–1547, 2011). We show that partial Bell polynomials are special cases of weighted integer compositions, and demonstrate how the identities for partial Bell polynomials easily follow from more general identities for weighted integer compositions. We also provide short and elegant probabilistic proofs of the latter, in terms of sums of discrete integer-valued random variables. Finally, we outline further identities for the partial Bell polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.


  1. Belbachir H., Bouroubi S., Khelladi A.: Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution. Ann. Math. Inform. 35, 21–30 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Cvijović D.: New identities for the partial Bell polynomials. Appl. Math. Lett. 24, 1544–1547 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. De Pril N.: Recursions for convolutions of arithmetic distributions. Astin Bull. 15, 135–139 (1985)

    Article  Google Scholar 

  4. Eger, S.: Restricted weighted integer compositions and extended binomial coefficients. J. Integer Seq. 16 (2013)

  5. Fahssi, N.-E.: The polynomial triangles revisited. Preprint available at (2012)

  6. Guo, Y.-H.: Some n-color compositions. J. Integer Seq. 15 (2012)

  7. Hoggatt V.E., Lind D.A.: Compositions and Fibonacci Numbers. Fibonacci Quart. 7, 253–266 (1969)

    MathSciNet  MATH  Google Scholar 

  8. Johnson W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Mon. 109, 217–234 (2002)

    Article  MATH  Google Scholar 

  9. Mansour T., Shattuck M.: A statistic on n-color compositions and related sequences. Proc. Indian Acad. Sci. (Math. Sci.) 124, 127–140 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shapcott C.: C-color compositions and palindromes. Fibonacci Quart. 50, 297–303 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Steffen Eger.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eger, S. Identities for partial Bell polynomials derived from identities for weighted integer compositions. Aequat. Math. 90, 299–306 (2016).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification