EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL Authored by: Hung-ping Tsao, Ph.D. 曹恆平 [email protected] and Leon Chang, Ph.D. 張法亷 [email protected] Tsao, Hung-ping and Chang, Leon (2021). Evolutionary mathematics and science for General famous numbers: Stirling-Euler-Lah-Bell. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 3, Number 1, January 2021; 99 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL3-NUM1- JAN2021; ISBN 978-0-9890870-3-2. US Department of Commerce, National Technical Information Service, 5301 Shawnee Road, Alexandria, VA 22312, USA. 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REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE NEW RESEARCH REPORT 3. DATES COVERED (From - To) 1-31-2021 JAN 2003-JAN 2021 4. TITLE AND SUBTITLE EVOLUTIONARY MATHEMATICS AND SCIENCE FOR 5a. CONTRACT NUMBER N/A GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL 5b. GRANT NUMBER N/A 5c. PROGRAM ELEMENT NUMBER N/A 6. AUTHOR(S) Tsao, Hung-ping and Chang,TECHNICAL Leon 5d. PROJECT NUMBER REPORT STEAM-VOL3-NUM1-JAN2021 5e. TASK NUMBER N/A 5f. WORK UNIT NUMBER N/A 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER STEAM-VOL3-NUM1-JAN2021 TSAO RESIDENCE, 1151 Highland Drive, Novato, CA 94949, USA 9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) Wang, Lawrence K. and Tsao, Hung-ping (editors). LENOX "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Volume 3, Number 1, January 2021; Lenox Institute 11. SPONSOR/MONITOR’S REPORT NUMBER(S) STEAM-VOL3-NUM1-JAN2021 Press, PO Box 405, Newtonville, NY, 12128-0405, USA 12. DISTRIBUTION / AVAILABILITY STATEMENT NO RESTRICTION 13. SUPPLEMENTARY NOTES REGISTRATION ISBN 978-0-9890870-3-2 푛 푛 14.TSAO ABSTRACT RESIDENCE, We first introduce 1151 Pascal, Highland Stirling, Eulerian, Drive Lah, Novato, and Bell numbers CA 94949,via sorting, USA then generalize Stirling numbers of both kinds [ ], { }, Eulerian numbers of 푘 푘 푛 푛 푛 푗 푛 푘 + 1 푛 two orders 〈 〉, 〈〈 〉〉, Lah numbers 퐿(푛, 푘) = ∑푛 [ ] { } and ∑푛−1 2푘 〈 〉 = ∑푛 (∑푘+1 [ ]) { }, the right-hand side of which is an ordered Bell polynomial, from the 푘 푘 푗=1 푗 푘 푘=0 푘 푘=1 푗−1 푗 푘 natural sequence based to arithmetically progressive sequences based. We further construct two types of arrays with any infinite sequence base A(i): T(n, k | A(i) | u;v), with T(0,0)=1, and u, v each indicating which weight to be used among W(1)=1, W(2)=A(n-1), W(3)=A(k), W(4)=A(n+k-1) +2A(1)-A(2), W(5)=A(n-k+1), W(6)=A(k+2)-2A(1) and W(7)=A(2n-k) in the recursive formula T(n, k)=W(u)T(n-1,k-1)+W(v)T(n-1,k). 1. Stirling: u=1, 0 ≤ 푘 ≤ 푛 and the initial values T(n, 0)=0 for n > 0. 푛 푛 푛 푛 푛 푛 [ ] =T(n, k | A(i)=a+(i-1)d | 1;2), [ ] = [ ], possessing diagonal (Stirling) polynomials; { } =T(n, k | A(i)=a+(i-1)d | 1;3), { } = { }, possessing diagonal 푘 푎;푑 푘 1;1 푘 푘 푎;푑 푘 1;1 푘 polynomials; Lad(n, k)=T(n, k | A(i)=a+(i-1)d | 1;4), L11(n, k)=L(n, k), possessing diagonal polynomials. 2. Eulerian: u>1, −1 ≤ 푘 ≤ 푛 − 1 and T(n, 0), n > 0, varies for each 푛 푛 array. 〈 〉 =T(n, k+1 | A(i)=a+(i-1)d | 5;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0; 〈〈 〉〉 =T(n, k+1 | A(i)=a+(i-1)d | 7;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0. 푘 푎;푑 푘 푎;푑 We shall introduce more Eulerian arrays with various initial values T(n, 0) for n > 0 and prove the existence of the general Stirling polynomials. Python programs are used to produce tables for both types of arrays along with difference tables of their diagonals to facilitate the calculation of diagonal polynomials in the case of existence. 15. SUBJECT TERMS (Keywords) Polynomial expression, Natural sequence, Binomial coefficient, Stirling number, Lah number, Natural sequence, Pascal triangle, Bernoulli coefficient, Arithmetically progressive sequence, Recursive formula, Sorting, Cycle, Subset, Eulerian number, Second generation Stirling numbers, Second generation Eulerian numbers, Stirling polynomial, Diagonal polynomial, q-Gaussian coefficient, Bell number, Ordered Bell polynomial, Python. 16. SECURITY CLASSIFICATION OF: UNCLASSIFIED UNLIMITED 17. LIMITATION 18. NUMBER OF 19a NAME OF RESPONSIBLE (UU) OF ABSTRACT UU PAGES 99 PERSON Wang, Lawrence K. a. REPORT UU b. ABSTRACT c. THIS PAGE 19a. TELEPHONE NUMBER (include area UU UU code) (518) 250-0012 Standard Form 298 (Rev. 8-98) Prescribed by ANSI St EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL Authored by: Hung-ping Tsao, Ph.D. 曹恆平 [email protected] and Leon Chang, Ph.D. 張法亷 [email protected] 3 TABLE OF CONTENTS TABLE OF CONTENTS ABSTRACT KEYWORDS NOMENCLATURE 1. INTRODUCTION 2. SORTING 3. RECURSION 4. PROPAGATION 5. GENERALIZTION 6. EXPLORATION 7. PYTHON PROGRAMS GLOSSARY REFERENCES APPENDIX A: LIST OF TABLES APPENDIX B: PYTHON PRODUCED TABLES EDITORS PAGE E-BOOK SERIES AND CHAPTER INTRODUCTON 4 EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL Hung-ping Tsao 曹恆平 and Leon Chang 張法亷 ABSTRACT We first introduce Pascal, Stirling, Eulerian, Lah and Bell numbers via sorting, then generalize Stirling numbers 푛 푛 푛 푛 푛 푗 of both kinds [ ], { }, Eulerian numbers of two orders 〈 〉, 〈〈 〉〉, Lah numbers 퐿(푛, 푘) = ∑푛 [ ] { } and 푘 푘 푘 푘 푗=1 푗 푘 푛 푘 + 1 푛 ∑푛−1 2푘 〈 〉 = ∑푛 (∑푘+1 [ ]) { }, the right-hand side of which is an ordered Bell polynomial, from the 푘=0 푘 푘=1 푗−1 푗 푘 natural sequence based to arithmetically progressive sequences based. We further construct two types of arrays with any infinite sequence base A(i): T(n, k | A(i) | u;v), with T(0,0)=1, and u, v each indicating which weight to be used among W(1)=1, W(2)=A(n-1), W(3)=A(k), W(4)=A(n+k-1) +2A(1)-A(2), W(5)=A(n-k+1), W(6)=A(k+2)-2A(1) and W(7)=A(2n-k) in the recursive formula T(n, k)=W(u)T(n-1,k-1)+W(v)T(n-1,k). 1. Stirling: u=1, 0 ≤ 푘 ≤ 푛 and the initial values T(n, 0)=0 for n > 0. 푛 푛 푛 [ ] =T(n, k | A(i)=a+(i-1)d | 1;2), [ ] = [ ], possessing diagonal (Stirling) polynomials; 푘 푎;푑 푘 1;1 푘 푛 푛 푛 { } =T(n, k | A(i)=a+(i-1)d | 1;3), { } = { }, possessing diagonal polynomials; 푘 푎;푑 푘 1;1 푘 Lad(n, k)=T(n, k | A(i)=a+(i-1)d | 1;4), L11(n, k)=L(n, k), possessing diagonal polynomials. 2. Eulerian: u>1, −1 ≤ 푘 ≤ 푛 − 1 and T(n, 0), n > 0, varies for each array. 푛 〈 〉 =T(n, k+1 | A(i)=a+(i-1)d | 5;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0; 푘 푎;푑 푛 〈〈 〉〉 =T(n, k+1 | A(i)=a+(i-1)d | 7;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0. 푘 푎;푑 We shall introduce more Eulerian arrays with various initial values T(n, 0) for n > 0 and prove the existence of the general Stirling polynomials. Python programs are used to produce tables for both types of arrays along with difference tables of their diagonals to facilitate the calculation of diagonal polynomials in the case of existence. Keywords: Natural sequence, Binomial coefficient, Stirling number, Lah number, Natural sequence, Pascal triangle, Bernoulli coefficient, Arithmetically progressive sequence, Recursive formula, Sorting, Cycle, Subset, Eulerian number, Second generation Stirling numbers, Second generation Eulerian numbers, Stirling polynomial, Diagonal polynomial, q-Gaussian coefficient, Bell number, Ordered Bell polynomial, Python. 5 NOMENCLATURE 푛 C(n, k), (푘) combination Σ sum the natural sequence (i)1 P(n, k) , the permutation of n elements taken k at a time k! k factorial L(n, k) Lah number n first-order Eulerian number k n second-order Eulerian number k n Stirling number of the first kind k n Stirling number of the second kind k (a (i 1)d)1 arithmetically progressive sequence n Stirling triangle of the first kind for k a;d n Stirling triangle of the second kind for k a;d n first-order Eulerian number for (a (i 1)d) 0 k a;d k second-order Eulerian number for j a;d r,s T (ai )0 general triangular array for (ai ) 0 ∏ product n1 q - Gaussian coefficient k q 퐵푛 Bell number 6 1.