[math-fun] Parenthesizing
Apropos of parenthesizing exponential towers, here's an easier question (probably not new): Look at all the (Catalan-many) ways to parenthesize a_1 - a_2 - ... - a_n. Not all are distinct as linear functions of a_1, a_2, ..., a_n; e.g., (a - (b - c)) - d = a - (b - (c - d)). How many different functions can be obtained? I don't know the answer. Is it 2^(n-2)? Jim Propp
I got my head in a little twist over this whole thread. Reassure me: do all the different parethesizations of 3^3^3^...^3, for n 3's, give different values? The original problem (using principle values of a^b) is interesting because i is special, right? On Wed, Nov 29, 2017 at 9:46 AM, James Propp <jamespropp@gmail.com> wrote:
Apropos of parenthesizing exponential towers, here's an easier question (probably not new): Look at all the (Catalan-many) ways to parenthesize a_1 - a_2 - ... - a_n. Not all are distinct as linear functions of a_1, a_2, ..., a_n; e.g.,
(a - (b - c)) - d = a - (b - (c - d)).
How many different functions can be obtained? I don't know the answer. Is it 2^(n-2)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The classic paper %D A000108 R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876. %H A000108 R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a> (annotated cached copy) has the answers (those two lines are from the OEIS entry for Catalan numbers, but the articles is referenced in many other entries) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Nov 29, 2017 at 10:24 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I got my head in a little twist over this whole thread. Reassure me: do all the different parethesizations of 3^3^3^...^3, for n 3's, give different values? The original problem (using principle values of a^b) is interesting because i is special, right?
On Wed, Nov 29, 2017 at 9:46 AM, James Propp <jamespropp@gmail.com> wrote:
Apropos of parenthesizing exponential towers, here's an easier question (probably not new): Look at all the (Catalan-many) ways to parenthesize a_1 - a_2 - ... - a_n. Not all are distinct as linear functions of a_1, a_2, ..., a_n; e.g.,
(a - (b - c)) - d = a - (b - (c - d)).
How many different functions can be obtained? I don't know the answer. Is it 2^(n-2)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Allan Wechsler -
James Propp -
Neil Sloane