### Abstract

Let Sing_{n} be the semigroup of singular self-maps of X_{n} = {1, ... ,n}, let R_{n} = {α ∈ Sing_{n}: (∀y ∈ Im α)|yα^{-1}| ≥ |Im α|} and let E(R_{n}) be the set of idempotents of R_{n}. Then it is shown that R_{n} = (E(R_{n}))^{2}. Moreover, expressions for the order of R_{n} and E(R_{n}) are obtained in terms of the kth-upper Stirling number of the second kind, S(n,r,k); defined as the number of partitions of X_{n} into r subsets each of size not less than k.

Original language | English |
---|---|

Pages (from-to) | 291-297 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 189 |

Issue number | 1-3 |

Publication status | Published - Jul 28 1998 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*189*(1-3), 291-297.

**Enumeration of certain finite semigroups of transformations.** / Umar, Abdullahi.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 189, no. 1-3, pp. 291-297.

}

TY - JOUR

T1 - Enumeration of certain finite semigroups of transformations

AU - Umar, Abdullahi

PY - 1998/7/28

Y1 - 1998/7/28

N2 - Let Singn be the semigroup of singular self-maps of Xn = {1, ... ,n}, let Rn = {α ∈ Singn: (∀y ∈ Im α)|yα-1| ≥ |Im α|} and let E(Rn) be the set of idempotents of Rn. Then it is shown that Rn = (E(Rn))2. Moreover, expressions for the order of Rn and E(Rn) are obtained in terms of the kth-upper Stirling number of the second kind, S(n,r,k); defined as the number of partitions of Xn into r subsets each of size not less than k.

AB - Let Singn be the semigroup of singular self-maps of Xn = {1, ... ,n}, let Rn = {α ∈ Singn: (∀y ∈ Im α)|yα-1| ≥ |Im α|} and let E(Rn) be the set of idempotents of Rn. Then it is shown that Rn = (E(Rn))2. Moreover, expressions for the order of Rn and E(Rn) are obtained in terms of the kth-upper Stirling number of the second kind, S(n,r,k); defined as the number of partitions of Xn into r subsets each of size not less than k.

UR - http://www.scopus.com/inward/record.url?scp=0042284272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042284272&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042284272

VL - 189

SP - 291

EP - 297

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -