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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(n) = \displaystyle {!}^{!n}
Bell numbers A000110(n) = \displaystyle {!}^{n}
Number of partitions of {1,...,n} A000262(n) = \displaystyle {!}^{n+}
Fubini numbers A000670(n) = \displaystyle {\Sigma}_{>}^{n}
E.g.f.: e^(2*(e^x - 1)) A001861(n) = \displaystyle {!}^{\underline{!} \cdot n}
Max_{k} { Number of partitions of n into k positive parts } A002569(n) = \displaystyle {!\max}_{>}^{!n}
n!*2^(n-1) A002866(n) = \displaystyle {\Sigma}_{>}^{n+}
(n+1)!*binomial(n,floor(n/2)) A002867(n-1) = \displaystyle {\max}_{>}^{n+}
Largest number in n-th row of triangle A008297 A002868(n) = \displaystyle {!\max}_{>}^{n+}
Largest number in n-th row of triangle A019538 A002869(n) = \displaystyle {\max}_{>}^{n}
Max_{k} Stirling2(n,k) A002870(n) = \displaystyle {!\max}_{>}^{n}
Max_{k} 2^k*Stirling2(n,k) A002871(n) = \displaystyle {!\max}_{>}^{\underline{!}\cdot n}
Column 2 of A162663 A002872(n) = \displaystyle {!}^{\underline{!}2\cdot n} = {!}^{\underline{cy}2\cdot n}
"Sorting numbers" A002873(n) = \displaystyle {!\max}_{>}^{\underline{!}2\cdot n}
Column 3 of A162663 A002874(n) = \displaystyle {!}^{\underline{cy}3\cdot n}
"Sorting numbers" A002875(n) = \displaystyle {!\max}_{>}^{\underline{cy}{3\cdot n}}
Stirling numbers of the second kind A008277(n,k) = \displaystyle {!k}_{>}^{n}
Falling factorial A008279(n,k) = \displaystyle {n}_{<}^{k}
Number of partitions of n into k positive parts A008284(n,k) = \displaystyle {!k}_{>}^{!n}
k!*Stirling2(n,k) A019538(n,k) = \displaystyle {k}_{>}^{n}
Number of partitions of n into at most k positive parts A026820(n,k) = \displaystyle {!k}^{!n}
Column 5 of A162663 A036075(n) = \displaystyle {!}^{\underline{cy}5\cdot n}
Column 7 of A162663 A036077(n) = \displaystyle {!}^{\underline{cy}7\cdot n}
Column 11 of A162663 A036081(n) = \displaystyle {!}^{\underline{cy}11\cdot n}
Sum_{i<=k} Stirling2(n,i) A102661(n,k) = \displaystyle {!k}^{n}
Column 13 of A162663 A141009(n) = \displaystyle {!}^{\underline{cy}13\cdot n}
n!*binomial(n-1,k-1) A156992(n,k) = \displaystyle {k}_{>}^{n+}

See also


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