Bell number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Bell number" in English language version.

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Halmos, Paul R. (1974). Naive set theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg. pp. 27–28. ISBN 9781475716450. MR 0453532.
Williams 1945 credits this observation to Silvio Minetola's Principii di Analisi Combinatoria (1909). Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Claesson (2001). Claesson, Anders (2001). "Generalized pattern avoidance". European Journal of Combinatorics. 22 (7): 961–971. arXiv:math/0011235. doi:10.1006/eujc.2001.0515. MR 1857258.
Callan (2006). Callan, David (2006). "A combinatorial interpretation of the eigensequence for composition". Journal of Integer Sequences. 9 (1): 06.1.4. arXiv:math/0507169. Bibcode:2005math......7169C. MR 2193154.
Rota 1964. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805.
Williams 1945. Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Wagstaff 1996. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the period of the Bell numbers modulo a prime". Mathematics of Computation. 65 (213): 383–391. Bibcode:1996MaCom..65..383W. doi:10.1090/S0025-5718-96-00683-7. MR 1325876.
Engel 1994. Engel, Konrad (1994). "On the average rank of an element in a filter of the partition lattice". Journal of Combinatorial Theory. Series A. 65 (1): 67–78. doi:10.1016/0097-3165(94)90038-8. MR 1255264.
Canfield 1995. Canfield, E. Rodney (1995). "Engel's inequality for Bell numbers". Journal of Combinatorial Theory. Series A. 72 (1): 184–187. doi:10.1016/0097-3165(95)90033-0. MR 1354972.
Asai, Kubo & Kuo 2000. Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung (2000). "Bell numbers, log-concavity, and log-convexity". Acta Applicandae Mathematicae. 63 (1–3): 79–87. arXiv:math/0104137. doi:10.1023/A:1010738827855. MR 1831247. S2CID 16533831.
Rota 1964. However, Rota gives an incorrect date, 1934, for Becker & Riordan 1948. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2): 385–394. doi:10.2307/2372336. JSTOR 2372336..

archive.org

Conway & Guy (1996). Conway, John Horton; Guy, Richard K. (1996). "Famous Families of Numbers: Bell Numbers and Stirling Numbers". The Book of Numbers. Copernicus Series. Springer. pp. 91–94. ISBN 9780387979939.
Flajolet & Sedgewick 2009. Flajolet, Philippe; Sedgewick, Robert (2009). "II.3 Surjections, set partitions, and words". Analytic Combinatorics. Cambridge University Press. pp. 106–119.
Dobiński 1877. Dobiński, G. (1877). "Summirung der Reihe $\textstyle \sum {\frac {n^{m}}{n!}}$ für m = 1, 2, 3, 4, 5, …". Grunert's Archiv. 61: 333–336.

arxiv.org

Claesson (2001). Claesson, Anders (2001). "Generalized pattern avoidance". European Journal of Combinatorics. 22 (7): 961–971. arXiv:math/0011235. doi:10.1006/eujc.2001.0515. MR 1857258.
Callan (2006). Callan, David (2006). "A combinatorial interpretation of the eigensequence for composition". Journal of Integer Sequences. 9 (1): 06.1.4. arXiv:math/0507169. Bibcode:2005math......7169C. MR 2193154.
Hurst & Schultz (2009). Hurst, Greg; Schultz, Andrew (2009). "An elementary (number theory) proof of Touchard's congruence". arXiv:0906.0696 [math.CO].
Asai, Kubo & Kuo 2000. Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung (2000). "Bell numbers, log-concavity, and log-convexity". Acta Applicandae Mathematicae. 63 (1–3): 79–87. arXiv:math/0104137. doi:10.1023/A:1010738827855. MR 1831247. S2CID 16533831.

asiapacific-mathnews.com

Gardner 1978 and Berndt 2011 also mention the connection between Bell numbers and The Tale of Genji, in less detail. Gardner, Martin (1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes". Scientific American. 238 (5): 24–30. Bibcode:1978SciAm.238e..24G. doi:10.1038/scientificamerican0578-24. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38 Berndt, Bruce C. (2011). "Ramanujan Reaches His Hand From His Grave To Snatch Your Theorems From You" (PDF). Asia Pacific Mathematics Newsletter. 1 (2): 8–13.
Berndt 2011. Berndt, Bruce C. (2011). "Ramanujan Reaches His Hand From His Grave To Snatch Your Theorems From You" (PDF). Asia Pacific Mathematics Newsletter. 1 (2): 8–13.

austinmohr.com

Canfield, Rod (July 1994). "The Moser-Wyman expansion of the Bell numbers" (PDF). Retrieved 2013-10-24.

books.google.com

Halmos, Paul R. (1974). Naive set theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg. pp. 27–28. ISBN 9781475716450. MR 0453532.
Lovász (1993). Lovász, L. (1993). "Section 1.14, Problem 9". Combinatorial Problems and Exercises (2nd ed.). Amsterdam, Netherlands: North-Holland. p. 17. ISBN 9780821869475. Zbl 0785.05001.

caltech.edu

math.caltech.edu

Simon, Barry (2010). "Example 15.4.6 (Asymptotics of Bell Numbers)". Complex Analysis (PDF). pp. 772–774. Archived from the original (PDF) on 2014-01-24. Retrieved 2012-09-02.

doi.org

Gardner 1978. Gardner, Martin (1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes". Scientific American. 238 (5): 24–30. Bibcode:1978SciAm.238e..24G. doi:10.1038/scientificamerican0578-24. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38
Williams 1945 credits this observation to Silvio Minetola's Principii di Analisi Combinatoria (1909). Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Claesson (2001). Claesson, Anders (2001). "Generalized pattern avoidance". European Journal of Combinatorics. 22 (7): 961–971. arXiv:math/0011235. doi:10.1006/eujc.2001.0515. MR 1857258.
Komatsu, Takao; Pita-Ruiz, Claudio (2018). "Some formulas for Bell numbers". Filomat. 32 (11): 3881–3889. doi:10.2298/FIL1811881K. ISSN 0354-5180.
Rota 1964. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805.
Becker & Riordan (1948). Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2): 385–394. doi:10.2307/2372336. JSTOR 2372336..
Williams 1945. Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Wagstaff 1996. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the period of the Bell numbers modulo a prime". Mathematics of Computation. 65 (213): 383–391. Bibcode:1996MaCom..65..383W. doi:10.1090/S0025-5718-96-00683-7. MR 1325876.
Engel 1994. Engel, Konrad (1994). "On the average rank of an element in a filter of the partition lattice". Journal of Combinatorial Theory. Series A. 65 (1): 67–78. doi:10.1016/0097-3165(94)90038-8. MR 1255264.
Canfield 1995. Canfield, E. Rodney (1995). "Engel's inequality for Bell numbers". Journal of Combinatorial Theory. Series A. 72 (1): 184–187. doi:10.1016/0097-3165(95)90033-0. MR 1354972.
Asai, Kubo & Kuo 2000. Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung (2000). "Bell numbers, log-concavity, and log-convexity". Acta Applicandae Mathematicae. 63 (1–3): 79–87. arXiv:math/0104137. doi:10.1023/A:1010738827855. MR 1831247. S2CID 16533831.
Bell 1934. Bell, E. T. (1934). "Exponential polynomials". Annals of Mathematics. 35 (2): 258–277. doi:10.2307/1968431. JSTOR 1968431..
Bell 1938. Bell, E. T. (1938). "The iterated exponential integers". Annals of Mathematics. 39 (3): 539–557. doi:10.2307/1968633. JSTOR 1968633..
Rota 1964. However, Rota gives an incorrect date, 1934, for Becker & Riordan 1948. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2): 385–394. doi:10.2307/2372336. JSTOR 2372336..
Gardner 1978 and Berndt 2011 also mention the connection between Bell numbers and The Tale of Genji, in less detail. Gardner, Martin (1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes". Scientific American. 238 (5): 24–30. Bibcode:1978SciAm.238e..24G. doi:10.1038/scientificamerican0578-24. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38 Berndt, Bruce C. (2011). "Ramanujan Reaches His Hand From His Grave To Snatch Your Theorems From You" (PDF). Asia Pacific Mathematics Newsletter. 1 (2): 8–13.

eudml.org

Callan (2006). Callan, David (2006). "A combinatorial interpretation of the eigensequence for composition". Journal of Integer Sequences. 9 (1): 06.1.4. arXiv:math/0507169. Bibcode:2005math......7169C. MR 2193154.

harvard.edu

ui.adsabs.harvard.edu

Gardner 1978. Gardner, Martin (1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes". Scientific American. 238 (5): 24–30. Bibcode:1978SciAm.238e..24G. doi:10.1038/scientificamerican0578-24. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38
Callan (2006). Callan, David (2006). "A combinatorial interpretation of the eigensequence for composition". Journal of Integer Sequences. 9 (1): 06.1.4. arXiv:math/0507169. Bibcode:2005math......7169C. MR 2193154.
Wagstaff 1996. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the period of the Bell numbers modulo a prime". Mathematics of Computation. 65 (213): 383–391. Bibcode:1996MaCom..65..383W. doi:10.1090/S0025-5718-96-00683-7. MR 1325876.
Gardner 1978 and Berndt 2011 also mention the connection between Bell numbers and The Tale of Genji, in less detail. Gardner, Martin (1978). "The Bells: versatile numbers that can count partitions of a set, primes and even rhymes". Scientific American. 238 (5): 24–30. Bibcode:1978SciAm.238e..24G. doi:10.1038/scientificamerican0578-24. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38 Berndt, Bruce C. (2011). "Ramanujan Reaches His Hand From His Grave To Snatch Your Theorems From You" (PDF). Asia Pacific Mathematics Newsletter. 1 (2): 8–13.

jstor.org

Williams 1945 credits this observation to Silvio Minetola's Principii di Analisi Combinatoria (1909). Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Rota 1964. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805.
Becker & Riordan (1948). Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2): 385–394. doi:10.2307/2372336. JSTOR 2372336..
Williams 1945. Williams, G. T. (1945). "Numbers generated by the function e^{e^x − 1}". American Mathematical Monthly. 52: 323–327. doi:10.2307/2305292. JSTOR 2305292. MR 0012612.
Bell 1934. Bell, E. T. (1934). "Exponential polynomials". Annals of Mathematics. 35 (2): 258–277. doi:10.2307/1968431. JSTOR 1968431..
Bell 1938. Bell, E. T. (1938). "The iterated exponential integers". Annals of Mathematics. 39 (3): 539–557. doi:10.2307/1968633. JSTOR 1968633..
Rota 1964. However, Rota gives an incorrect date, 1934, for Becker & Riordan 1948. Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504. doi:10.2307/2312585. JSTOR 2312585. MR 0161805. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2): 385–394. doi:10.2307/2372336. JSTOR 2372336..

nb.rs

doiserbia.nb.rs

Komatsu, Takao; Pita-Ruiz, Claudio (2018). "Some formulas for Bell numbers". Filomat. 32 (11): 3881–3889. doi:10.2298/FIL1811881K. ISSN 0354-5180.

oeis.org

Sloane, N. J. A. (ed.). "Sequence A011971 (Aitken's array)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A051131". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

purdue.edu

homes.cerias.purdue.edu

Wagstaff 1996. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the period of the Bell numbers modulo a prime". Mathematics of Computation. 65 (213): 383–391. Bibcode:1996MaCom..65..383W. doi:10.1090/S0025-5718-96-00683-7. MR 1325876.

semanticscholar.org

api.semanticscholar.org

Asai, Kubo & Kuo 2000. Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung (2000). "Bell numbers, log-concavity, and log-convexity". Acta Applicandae Mathematicae. 63 (1–3): 79–87. arXiv:math/0104137. doi:10.1023/A:1010738827855. MR 1831247. S2CID 16533831.

ucsd.edu

math.ucsd.edu

Bender & Williamson 2006. Bender, Edward A.; Williamson, S. Gill (2006). "Example 11.7, Set Partitions". Foundations of Combinatorics with Applications (PDF). Dover. pp. 319–320. ISBN 0-486-44603-4.

upenn.edu

math.upenn.edu

Wilf 1994, p. 23. Wilf, Herbert S. (1994). Generatingfunctionology (PDF) (2nd ed.). Boston, MA: Academic Press. ISBN 0-12-751956-4. Zbl 0831.05001.
Wilf 1994, pp. 20–23. Wilf, Herbert S. (1994). Generatingfunctionology (PDF) (2nd ed.). Boston, MA: Academic Press. ISBN 0-12-751956-4. Zbl 0831.05001.

web.archive.org

Simon, Barry (2010). "Example 15.4.6 (Asymptotics of Bell Numbers)". Complex Analysis (PDF). pp. 772–774. Archived from the original (PDF) on 2014-01-24. Retrieved 2012-09-02.

worldcat.org

Komatsu, Takao; Pita-Ruiz, Claudio (2018). "Some formulas for Bell numbers". Filomat. 32 (11): 3881–3889. doi:10.2298/FIL1811881K. ISSN 0354-5180.

zbmath.org

Wilf 1994, p. 23. Wilf, Herbert S. (1994). Generatingfunctionology (PDF) (2nd ed.). Boston, MA: Academic Press. ISBN 0-12-751956-4. Zbl 0831.05001.
Wilf 1994, pp. 20–23. Wilf, Herbert S. (1994). Generatingfunctionology (PDF) (2nd ed.). Boston, MA: Academic Press. ISBN 0-12-751956-4. Zbl 0831.05001.
Lovász (1993). Lovász, L. (1993). "Section 1.14, Problem 9". Combinatorial Problems and Exercises (2nd ed.). Amsterdam, Netherlands: North-Holland. p. 17. ISBN 9780821869475. Zbl 0785.05001.