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# Sorting numbers

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

## Sequence Notation Correspondence

Sequences of "Sorting Numbers" in the OEIS
Sequence Motzkin's Notation
Partition numbers A000041${\displaystyle (n)=\displaystyle {!}^{!n}}$
Bell numbers A000110${\displaystyle (n)=\displaystyle {!}^{n}}$
Number of partitions of {1,...,n} A000262${\displaystyle (n)=\displaystyle {!}^{n+}}$
Fubini numbers A000670${\displaystyle (n)=\displaystyle {\Sigma }_{>}^{n}}$
E.g.f.: e^(2*(e^x - 1)) A001861${\displaystyle (n)=\displaystyle {!}^{{\underline {!}}\cdot n}}$
Max_{k} { Number of partitions of n into k positive parts } A002569${\displaystyle (n)=\displaystyle {!\max }_{>}^{!n}}$
n!*2^(n-1) A002866${\displaystyle (n)=\displaystyle {\Sigma }_{>}^{n+}}$
(n+1)!*binomial(n,floor(n/2)) A002867${\displaystyle (n-1)=\displaystyle {\max }_{>}^{n+}}$
Largest number in n-th row of triangle A008297 A002868${\displaystyle (n)=\displaystyle {!\max }_{>}^{n+}}$
Largest number in n-th row of triangle A019538 A002869${\displaystyle (n)=\displaystyle {\max }_{>}^{n}}$
Max_{k} Stirling2(n,k) A002870${\displaystyle (n)=\displaystyle {!\max }_{>}^{n}}$
Max_{k} 2^k*Stirling2(n,k) A002871${\displaystyle (n)=\displaystyle {!\max }_{>}^{{\underline {!}}\cdot n}}$
Column 2 of A162663 A002872${\displaystyle (n)=\displaystyle {!}^{{\underline {!}}2\cdot n}={!}^{{\underline {cy}}2\cdot n}}$
Max_{k} #{partitions of 2n into k parts
which are invariant under (12)(34)...(2n-1,2n)}
A002873${\displaystyle (n)=\displaystyle {!\max }_{>}^{{\underline {!}}2\cdot n}}$
Column 3 of A162663 A002874${\displaystyle (n)=\displaystyle {!}^{{\underline {cy}}3\cdot n}}$
? A002875${\displaystyle (n)=\displaystyle {!\max }_{>}^{{\underline {cy}}{3\cdot n}}}$
Stirling numbers of the second kind A008277${\displaystyle (n,k)=\displaystyle {!k}_{>}^{n}}$
Falling factorial A008279${\displaystyle (n,k)=\displaystyle {n}_{<}^{k}}$
Number of partitions of n into k positive parts A008284${\displaystyle (n,k)=\displaystyle {!k}_{>}^{!n}}$
k!*Stirling2(n,k) A019538${\displaystyle (n,k)=\displaystyle {k}_{>}^{n}}$
Number of partitions of n into at most k positive parts A026820${\displaystyle (n,k)=\displaystyle {!k}^{!n}}$
Column 5 of A162663 A036075${\displaystyle (n)=\displaystyle {!}^{{\underline {cy}}5\cdot n}}$
Column 7 of A162663 A036077${\displaystyle (n)=\displaystyle {!}^{{\underline {cy}}7\cdot n}}$
Column 11 of A162663 A036081${\displaystyle (n)=\displaystyle {!}^{{\underline {cy}}11\cdot n}}$
Sum_{i<=k} Stirling2(n,i) A102661${\displaystyle (n,k)=\displaystyle {!k}^{n}}$
Column 13 of A162663 A141009${\displaystyle (n)=\displaystyle {!}^{{\underline {cy}}13\cdot n}}$
n!*binomial(n-1,k-1) A156992${\displaystyle (n,k)=\displaystyle {k}_{>}^{n+}}$