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Sorting numbers
From OeisWiki
This is the 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
| Sequence | Motzkin's Notation |
|---|---|
| Partition numbers | A000041Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{!n}} |
| Bell numbers | A000110Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{n}} |
| Number of partitions of {1,...,n} | A000262Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{n+}} |
| Fubini numbers | A000670Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {\Sigma}_{>}^{n}} |
| E.g.f.: e^(2*(e^x - 1)) | A001861Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{!} \cdot n}} |
| Max_{k} { Number of partitions of n into k positive parts } | A002569Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{!n}} |
| n!*2^(n-1) | A002866Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {\Sigma}_{>}^{n+}} |
| (n+1)!*binomial(n,floor(n/2)) | A002867Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n-1) = \displaystyle {\max}_{>}^{n+}} |
| Largest number in n-th row of triangle A008297 | A002868Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{n+}} |
| Largest number in n-th row of triangle A019538 | A002869Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {\max}_{>}^{n}} |
| Max_{k} Stirling2(n,k) | A002870Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{n}} |
| Max_{k} 2^k*Stirling2(n,k) | A002871Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{\underline{!}\cdot n}} |
| Column 2 of A162663 | A002872Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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)} |
A002873Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{\underline{!}2\cdot n}} |
| Column 3 of A162663 | A002874Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{cy}3\cdot n}} |
| ? | A002875Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!\max}_{>}^{\underline{cy}{3\cdot n}}} |
| Stirling numbers of the second kind | A008277Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {!k}_{>}^{n}} |
| Falling factorial | A008279Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {n}_{<}^{k}} |
| Number of partitions of n into k positive parts | A008284Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {!k}_{>}^{!n}} |
| k!*Stirling2(n,k) | A019538Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {k}_{>}^{n}} |
| Number of partitions of n into at most k positive parts | A026820Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {!k}^{!n}} |
| Column 5 of A162663 | A036075Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{cy}5\cdot n}} |
| Column 7 of A162663 | A036077Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{cy}7\cdot n}} |
| Column 11 of A162663 | A036081Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{cy}11\cdot n}} |
| Sum_{i<=k} Stirling2(n,i) | A102661Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {!k}^{n}} |
| Column 13 of A162663 | A141009Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n) = \displaystyle {!}^{\underline{cy}13\cdot n}} |
| n!*binomial(n-1,k-1) | A156992Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n,k) = \displaystyle {k}_{>}^{n+}} |
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]
