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# Sorting numbers

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The purpose of this article is to clarify the definitions and notation used by Motzkin in "Sorting numbers for cylinders and other classification numbers" (1971).

# Sequence Notation Coorespondence

Sequence | Motzkin's Notation |
---|---|

Partition numbers | A000041 |

Bell numbers | A000110 |

Number of partitions of {1,...,n} | A000262 |

Fubini numbers | A000670 |

E.g.f.: e^(2*(e^x - 1)) | A001861 |

Max_{k} { Number of partitions of n into k positive parts } | A002569 |

n!*2^(n-1) | A002866 |

(n+1)!*binomial(n,floor(n/2)) | A002867 |

Largest number in n-th row of triangle A008297 | A002868 |

Largest number in n-th row of triangle A019538 | A002869 |

Max_{k} Stirling2(n,k) | A002870 |

Max_{k} 2^k*Stirling2(n,k) | A002871 |

Column 2 of A162663 | A002872 |

"Sorting numbers" | A002873 |

Column 3 of A162663 | A002874 |

"Sorting numbers" | A002875 |

Stirling numbers of the second kind | A008277 |

Falling factorial | A008279 |

Number of partitions of n into k positive parts | A008284 |

k!*Stirling2(n,k) | A019538 |

Number of partitions of n into at most k positive parts | A026820 |

Column 5 of A162663 | A036075 |

Column 7 of A162663 | A036077 |

Column 11 of A162663 | A036081 |

Sum_{i<=k} Stirling2(n,i) | A102661 |

Column 13 of A162663 | A141009 |

n!*binomial(n-1,k-1) | A156992 |

## See also

## References

- T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]