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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

Sequences of "Sorting Numbers" in the OEIS
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