Timeline for Alternative proofs sought after for a certain identity

8 events

when toggle format	what		by	license	comment
Nov 2, 2021 at 16:32	history	edited	Ira Gessel	CC BY-SA 4.0	deleted 6 characters in body
Dec 21, 2020 at 19:44	comment	added	Iosif Pinelis		@PeterLeFanuLumsdaine : I agree with your comment.
Dec 21, 2020 at 16:16	comment	added	Peter LeFanu Lumsdaine		@IosifPinelis: I’d agree, in some sense this is essentially the same as the OP’s induction proof. But at the same time, this presentation is clearer and more informative in several ways — e.g. this version points the way more clearly to combinatorial/probabilistic interpretations such as that in Fedor Petrov’s answer.
Dec 18, 2020 at 21:07	comment	added	LSpice		MathJax note: multi-line-equation AMS environments like `{align*}` work here, and display better than a collection of inline equations separated by blank lines. I have edited accordingly.
Dec 18, 2020 at 21:06	history	edited	LSpice	CC BY-SA 4.0	{align*}
Dec 18, 2020 at 18:51	comment	added	Iosif Pinelis		To me, this looks like a concise way to rewrite the induction proof in the OP. In particular, the last equality in your display is essentially the same as the identity $\binom nj-\binom{n-1}j=\binom{n-1}{j-1}$, used in the beginning of the induction proof, whereas your first equality is essentially what was used at the end of the the induction proof.
Dec 18, 2020 at 18:19	history	edited	Alapan Das	CC BY-SA 4.0	deleted 10 characters in body
Dec 18, 2020 at 18:13	history	answered	Alapan Das	CC BY-SA 4.0