# On the 2-adic order of Stirling numbers of the second kind and their differences

Abstract : Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.
Keywords :
Type de document :
Communication dans un congrès
Krattenthaler, Christian and Strehl, Volker and Kauers, Manuel. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), pp.561-572, 2009, DMTCS Proceedings
Domaine :
Liste complète des métadonnées

Littérature citée [7 références]

https://hal.inria.fr/hal-01185386
Contributeur : Coordination Episciences Iam <>
Soumis le : jeudi 20 août 2015 - 11:07:02
Dernière modification le : mardi 7 mars 2017 - 15:04:45
Document(s) archivé(s) le : mercredi 26 avril 2017 - 10:10:39

### Fichier

dmAK0147.pdf
Fichiers éditeurs autorisés sur une archive ouverte

### Identifiants

• HAL Id : hal-01185386, version 1

### Citation

Tamás Lengyel. On the 2-adic order of Stirling numbers of the second kind and their differences. Krattenthaler, Christian and Strehl, Volker and Kauers, Manuel. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), pp.561-572, 2009, DMTCS Proceedings. 〈hal-01185386〉

### Métriques

Consultations de la notice

## 35

Téléchargements de fichiers