naa a22 4500 510784593 CHVBK 20180411083258.0 cr unu---uuuuu 180411e20130701xx s 000 0 eng 10.1007/s12045-013-0082-7 doi (NATIONALLICENCE)springer-10.1007/s12045-013-0082-7 Shirali Shailesh Sahyadri School, 410 513, Tiwai Hill, Rajgurunagar, Pune, India aut Combinatorial proofs and algebraic proofs — I [Elektronische Daten] [Shailesh Shirali] It is sometimes the case in mathematics that the same result can be proved in two or more essentially different ways (a luxury of mathematical riches, you might say). A recurrent theme in this context is the tension between combinatorial proofs and algebraic proofs. Inthis two-part article we describe some results which allow for proofs of both kinds and give us an opportunity to contrast them. We shall see some beautiful examples of both kinds of proofs. In Part I we dwell on results that deal with the binomial coefficients and with Fibonacci numbers. We will not be overly zealous in our usage of the term ‘combinatorial': we will allow a proof to be so categorized if it is largely combinatorial, with only a ‘small' component of algebra. However, that small part may well be an essential component of the proof. The simplicity of counting arguments can be deceptive. In recent times, Paul Erdős has shown just how subtle and powerful such arguments can be, and how far they can reach. Indian Academy of Sciences, 2013 Combinatorial proof nationallicence algebraic proof nationallicence binomial identity nationallicence recurrence relation nationallicence composition nationallicence Fibonacci number nationallicence Fibonacci identity nationallicence Pascal triangle nationallicence Resonance Springer India 18/7(2013-07-01), 630-645 0971-8044 18:7<630 2013 18 12045 https://doi.org/10.1007/s12045-013-0082-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s12045-013-0082-7 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Shirali Shailesh Sahyadri School, 410 513, Tiwai Hill, Rajgurunagar, Pune, India aut NATIONALLICENCE 773 0- Resonance Springer India 18/7(2013-07-01), 630-645 0971-8044 18:7<630 2013 18 12045 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer