On 2017-11-03 18:09, Neil Sloane wrote:
Quicker to look it up! https://oeis.org/A051288.
Duh, a month in the laboratory can often save an afternoon in the library.
I added the socksual interpretation.
Wow, I had never connected socksuality with foot-fetishism. But (sniff), all my socks are achiral, in more ways than one! "Think socks (those weird "chiral" kind.)" --rwg
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 Fri, Nov 3, 2017 at 8:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
Our old friend STAN.K bought n pairs of white "chiral" (podal!) socks clearly stamped L and R, which his housekeeper nevertheless paired willy-nilly. Stan wondered, for n pairs, how likely were 0, 2, 4, ... mismatches. I couldn't remember how to map a washer-dryer onto an urn, so I just tabulated
2
4 2
8 12
16 48 6
32 160 60
64 480 360 20
128 1344 1680 280
256 3584 6720 2240 70
512 9216 24192 13440 1260
and fit a formula with Mathematica. The nth row represents n pairs, row sum = binomial(2n,2). Thus the 2nd row, 4 2, represents 2 pairs, LRLR,LRRL,RLLR,RLRL = 4 perfectos, and LLRR,RRLL = 2 mismatches. (The left column is how many perfectos for n pairs.) The empirical formula I got was StringReverse@"]j2-n,j,j[laimonitluM)j2-n(^2" which then Julian promptly derived in a single line:
StringReverse@".LR dna RL neewteb esoohc ot syaw )j2-n(^2 eht yb deilpitlum neht si hcihw ,)!)j2-n(*2^!j(/!n si LR ro RL rehtie eb ot j2-n dna ,RR eb ot j ,LL eb ot sriap j esoohc ot syaw fo rebmun eht"
I guess that makes it fairly clear why the left (perfecto) column is 2^n. And, why the right diagonal is 2n choose n. --rwg