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
Quicker to look it up! https://oeis.org/A051288. I added the socksual interpretation. 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Heh, that'll learn 'em! Nit: "...neither 0,1 not 1,0 (think "socks")..." "not" should be "nor" Warm regards, --MLB
On Nov 3, 2017, at 6:09 PM, Neil Sloane <njasloane@gmail.com> wrote:
Quicker to look it up! https://oeis.org/A051288.
I added the socksual interpretation.
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 _______________________________________________ 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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Bill Gosper -
Marc LeBrun -
Neil Sloane