Bell number (English Wikipedia)

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

refsWebsite

Global rank English rank

15doi.org

2^nd place

11ams.org

451^st place

277^th place

7jstor.org

26^th place

20^th place

4harvard.edu

18^th place

17^th place

4arxiv.org

69^th place

59^th place

3zbmath.org

1,923^rd place

1,068^th place

3archive.org

6^th place

2books.google.com

3^rd place

2oeis.org

742^nd place

538^th place

2upenn.edu

702^nd place

520^th place

2asiapacific-mathnews.com

low place

1eudml.org

low place

1openmathbooks.org

low place

1nb.rs

3,800^th place

8,978^th place

1worldcat.org

5^th place

1ucsd.edu

1,933^rd place

1,342^nd place

1purdue.edu

850^th place

625^th place

1web.archive.org

1^st place

1caltech.edu

887^th place

714^th place

1semanticscholar.org

11^th place

8^th place

1austinmohr.com

low place

ams.org (Global: 451^st place; English: 277^th place)

mathscinet.ams.org

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 (Global: 6^th place; English: 6^th place)

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 (Global: 69^th place; English: 59^th place)

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 (Global: low place; English: low place)

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 (Global: low place; English: low place)

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

books.google.com (Global: 3^rd place; English: 3^rd place)

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 (Global: 887^th place; English: 714^th place)

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 (Global: 2^nd place; English: 2^nd place)

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 (Global: low place; English: low place)

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 (Global: 18^th place; English: 17^th place)

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 (Global: 26^th place; English: 20^th place)

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 (Global: 3,800^th place; English: 8,978^th place)

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 (Global: 742^nd place; English: 538^th place)

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.

openmathbooks.org (Global: low place; English: low place)

discrete.openmathbooks.org

"Stirling Numbers of the Second Kind, Theorem 3.4.1".

purdue.edu (Global: 850^th place; English: 625^th place)

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 (Global: 11^th place; English: 8^th place)

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 (Global: 1,933^rd place; English: 1,342^nd place)

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 (Global: 702^nd place; English: 520^th place)

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 (Global: 1^st place; English: 1^st place)

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 (Global: 5^th place; English: 5^th place)

search.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 (Global: 1,923^rd place; English: 1,068^th place)

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.

Bell number (English Wikipedia)

ams.org (Global: 451st place; English: 277th place)

mathscinet.ams.org

archive.org (Global: 6th place; English: 6th place)

arxiv.org (Global: 69th place; English: 59th place)

asiapacific-mathnews.com (Global: low place; English: low place)

austinmohr.com (Global: low place; English: low place)

books.google.com (Global: 3rd place; English: 3rd place)

caltech.edu (Global: 887th place; English: 714th place)

math.caltech.edu

doi.org (Global: 2nd place; English: 2nd place)

eudml.org (Global: low place; English: low place)

harvard.edu (Global: 18th place; English: 17th place)

ui.adsabs.harvard.edu

jstor.org (Global: 26th place; English: 20th place)

nb.rs (Global: 3,800th place; English: 8,978th place)

doiserbia.nb.rs

oeis.org (Global: 742nd place; English: 538th place)

openmathbooks.org (Global: low place; English: low place)

discrete.openmathbooks.org

purdue.edu (Global: 850th place; English: 625th place)

homes.cerias.purdue.edu

semanticscholar.org (Global: 11th place; English: 8th place)

api.semanticscholar.org

ucsd.edu (Global: 1,933rd place; English: 1,342nd place)

math.ucsd.edu

upenn.edu (Global: 702nd place; English: 520th place)

math.upenn.edu

web.archive.org (Global: 1st place; English: 1st place)

worldcat.org (Global: 5th place; English: 5th place)

search.worldcat.org

zbmath.org (Global: 1,923rd place; English: 1,068th place)

ams.org (Global: 451^st place; English: 277^th place)

archive.org (Global: 6^th place; English: 6^th place)

arxiv.org (Global: 69^th place; English: 59^th place)

books.google.com (Global: 3^rd place; English: 3^rd place)

caltech.edu (Global: 887^th place; English: 714^th place)

doi.org (Global: 2^nd place; English: 2^nd place)

harvard.edu (Global: 18^th place; English: 17^th place)

jstor.org (Global: 26^th place; English: 20^th place)

nb.rs (Global: 3,800^th place; English: 8,978^th place)

oeis.org (Global: 742^nd place; English: 538^th place)

purdue.edu (Global: 850^th place; English: 625^th place)

semanticscholar.org (Global: 11^th place; English: 8^th place)

ucsd.edu (Global: 1,933^rd place; English: 1,342^nd place)

upenn.edu (Global: 702^nd place; English: 520^th place)

web.archive.org (Global: 1^st place; English: 1^st place)

worldcat.org (Global: 5^th place; English: 5^th place)

zbmath.org (Global: 1,923^rd place; English: 1,068^th place)