January 2017 - mathfun@mailman.xmission.com

[math-fun] booltools
by Bill Gosper 17 Jan '17

17 Jan '17

Rohan requested all 7581 monotonic pentadic boolean functions. Instead of reading up on the smart way to do this (Vol IV?), I built a little sandbox, iteratively Outer Anding and Outer Oring the initial pool {0,p,q,r,s,t,1} with itself. Interestingly, if one Outerand is always {0,p,q,r,s,t,1}, this only finds 7580. The missing one is easy to guess. For brevity, we illustrate with the triadics: In[384]:= three = alg2bool /@ {0, p, q, r, 1} Out[384]= {(#1[[1]]; False) &, #1[[1]] &, #1[[2]] &, #1[[3]] &, (#1[[1]]; True) &} (For philosophical reasons, the constants True and False are monadic (vs nulladic.) What is the truth table of a constant?) ttab is truth table: In[385]:= ttab /@ three During evaluation of In[385]:= Transpose::nmtx: The first two levels of {} cannot be transposed. (A harmless glitch.) Out[385]= {{False}, {False, True}, {False, True, False, True}, {False, True, False, True, False, True, False, True}, {True}} In[386]:= three; cull is just delete duplicates by truth table. tim prints time and length. In[387]:= cull[Union[% // tim, cull[Outer[Distribute[And[#1, #2], Function] &, three, %] // Flatten // tim] // tim, cull[Outer[Distribute[Or[#1, #2], Function] &, three, %] // Flatten // tim] // tim]] // tim During evaluation of In[387]:= 6.*10^-6,5 (6 μs, 5 fcns) During evaluation of In[387]:= 0.000064,25 During evaluation of In[387]:= 0.015481,8 During evaluation of In[387]:= 0.000063,25 During evaluation of In[387]:= 0.002756,8 During evaluation of In[387]:= 0.020625,11 (Redundantly culling smaller pieces saves time.) Out[387]= {(#1[[1]]; False) && (#1[[1]]; False) &, (#1[[1]]; True) && (#1[[1]]; True) &, #1[[1]] && #1[[1]] &, #1[[1]] && #1[[ 2]] &, #1[[1]] && #1[[3]] &, #1[[2]] && #1[[2]] &, #1[[2]] && #1[[ 3]] &, #1[[3]] && #1[[3]] &, #1[[1]] || #1[[2]] &, #1[[1]] || #1[[ 3]] &, #1[[2]] || #1[[3]] &} This brought us up to 11 functions. Rub these against the original 5: In[388]:= cull[Union[% // tim, cull[Outer[Distribute[And[#1, #2], Function] &, three, %] // Flatten // tim] // tim, cull[Outer[Distribute[Or[#1, #2], Function] &, three, %] // Flatten // tim] // tim]] // tim During evaluation of In[388]:= 0.000013,11 During evaluation of In[388]:= 0.000124,55 During evaluation of In[388]:= 0.031731,15 During evaluation of In[388]:= 0.00013,55 During evaluation of In[388]:= 0.007292,15 During evaluation of In[388]:= 0.044341,19 Out[388]= {(#1[[1]]; False) && ((#1[[1]]; False) && (#1[[1]]; False)) &, (#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]]; True)) &, (#1[[1]]; True) && (#1[[1]] || #1[[2]]) &, (#1[[1]]; True) && (#1[[1]] || #1[[3]]) &, (#1[[1]]; True) && (#1[[2]] || #1[[3]]) &, #1[[ 1]] && ((#1[[1]]; True) && (#1[[1]]; True)) &, #1[[ 1]] && (#1[[1]] && #1[[2]]) &, #1[[ 1]] && (#1[[1]] && #1[[3]]) &, #1[[ 1]] && (#1[[2]] && #1[[3]]) &, #1[[ 1]] && (#1[[2]] || #1[[3]]) &, #1[[ 2]] && ((#1[[1]]; True) && (#1[[1]]; True)) &, #1[[ 2]] && (#1[[2]] && #1[[3]]) &, #1[[ 2]] && (#1[[1]] || #1[[3]]) &, #1[[ 3]] && ((#1[[1]]; True) && (#1[[1]]; True)) &, #1[[ 3]] && (#1[[1]] || #1[[2]]) &, #1[[ 1]] || (#1[[2]] && #1[[3]]) &, #1[[ 1]] || (#1[[2]] || #1[[3]]) &, #1[[ 2]] || (#1[[1]] && #1[[3]]) &, #1[[3]] || (#1[[1]] && #1[[2]]) &} We got 19 of the 20. But when we try again, In[389]:= cull[Union[% // tim, cull[Outer[Distribute[And[#1, #2], Function] &, three, %] // Flatten // tim] // tim, cull[Outer[Distribute[Or[#1, #2], Function] &, three, %] // Flatten // tim] // tim]] // tim During evaluation of In[389]:= 0.000017,19 During evaluation of In[389]:= 0.000214,95 During evaluation of In[389]:= 0.063244,19 During evaluation of In[389]:= 0.000211,95 During evaluation of In[389]:= 0.016854,19 During evaluation of In[389]:= 0.090048,19 . . . we stick at 19. The solution is to Outer with the first iterate, In[458]:= bool2alg /@ %387 Out[458]= {0, 1, p, p q, p r, q, q r, r, p + q, p + r, q + r} rather than {0, p, q, r, 1}, the 0th. In[390]:= cull[Union[% // tim, cull[Outer[Distribute[And[#1, #2], Function] &, %387, %] // Flatten // tim] // tim, cull[Outer[Distribute[Or[#1, #2], Function] &, %387, %] // Flatten // tim] // tim]] // tim During evaluation of In[390]:= 0.000018,19 During evaluation of In[390]:= 0.000466,209 During evaluation of In[390]:= 0.200324,20 During evaluation of In[390]:= 0.00059,209 During evaluation of In[390]:= 0.037646,20 During evaluation of In[390]:= 0.248116,20 Out[390]= {((#1[[1]]; False) && (#1[[1]]; False)) && ((#1[[1]]; False) && ((#1[[1]]; False) && ((#1[[1]]; False) && (#1[[1]]; False)))) &, ((#1[[ 1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]]; True)))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]] || #1[[2]]))) &, ((#1[[ 1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]] || #1[[3]]))) &, ((#1[[ 1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[2]] || #1[[3]]))) &, ((#1[[ 1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && (#1[[1]] || (#1[[2]] && #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && (#1[[1]] || (#1[[2]] || #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && (#1[[2]] || (#1[[1]] && #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && ((#1[[1]]; True) && (#1[[3]] || (#1[[1]] && #1[[2]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 1]] && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]]; True)))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 1]] && ((#1[[1]]; True) && (#1[[2]] || #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 1]] && (#1[[1]] && (#1[[1]] && #1[[2]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 1]] && (#1[[1]] && (#1[[1]] && #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 1]] && (#1[[1]] && (#1[[2]] && #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 2]] && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]]; True)))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 2]] && ((#1[[1]]; True) && (#1[[1]] || #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 2]] && (#1[[2]] && (#1[[2]] && #1[[3]]))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 3]] && ((#1[[1]]; True) && ((#1[[1]]; True) && (#1[[1]]; True)))) &, ((#1[[1]]; True) && (#1[[1]]; True)) && (#1[[ 3]] && ((#1[[1]]; True) && (#1[[1]] || #1[[2]]))) &, (#1[[ 1]] || #1[[2]]) && ((#1[[1]]; True) && (#1[[3]] || (#1[[1]] && #1[[2]]))) &} The last one: In[391]:= bool2alg@%[[-1]] Out[391]= (p + q) (p q + r) is not obviously symmetric. In[392]:= ttab@%%[[-1]] == ttab@alg2bool[p q + p r + q r] Out[392]= True Note that this function is insensitive to swapping And with Or: In[459]:= p q + p r + q r /. {Times -> Plus, Plus -> Times} Out[459]= (p + q) (p + r) (q + r) In[460]:= bool2alg@alg2bool@Expand@% Out[460]= p q + p r + q r + p q r (I didn't write a simplifier, but the p q r is clearly redundant.) The 7581st function was (p + q + r) (p + q + s) (p + r + s) (q + r + s) (p + q + t) (p + r + t) (q + r + t) (p + s + t) (q + s + t) (r + s + t) == p q r + p q s + p r s + q r s + p q t + p r t + q r t + p s t + q s t + r s t While guessing and testing this, the all-day (46000 second) cull finished. But I was a couple of minutes bringing up the window, and the PoS OSX decided to reboot. This was almost certainly not a hardware error. Tom Knight told me Microsoft and Apple don't even try to makes stable environments. Symbolics Genera stayed up months at a time. If anyone from Symbolics offered to teach the big guys how to make stable systems, they would have been shown the door. But I bet Google knows how to do it. Code: In[376]:= Clear@bool2alg; bool2alg[bool_] := (bool /. {False -> 0, True -> 1, And -> Times, Or -> Plus})@{p, q, r, s, t} /. x_^_ :> x In[377]:= five = {(#[[1]]; False) &, #[[1]] &, #[[2]] &, #[[3]] &, #[[ 4]] &, #[[5]] &, (#[[1]]; True) &} Out[377]= {(#1[[1]]; False) &, #1[[1]] &, #1[[2]] &, #1[[3]] &, #1[[ 4]] &, #1[[5]] &, (#1[[1]]; True) &} In[378]:= bool2alg /@ % Out[378]= {0, p, q, r, s, t, 1} In[448]:= Clear@alg2bool; alg2bool[exp_] := exp //. Except[0, _Integer] :> 1 /. { 0 -> ((#1[[1]]; False) &), 1 -> ((#1[[1]]; True) &)} /. S : (p | q | r | s | t) :> (#[[x]] & /. x :> Evaluate@(Position[{p, q, r, s, t}, S][[1, 1]])) /. {Times -> And, Plus -> Or} //. f : (And | Or)[Function[_] ..] :> Distribute[f, Function] In[380]:= alg2bool /@ %% Out[380]= {(#1[[1]]; False) &, #1[[1]] &, #1[[2]] &, #1[[3]] &, #1[[ 4]] &, #1[[5]] &, (#1[[1]]; True) &} In[381]:= Clear@ttab; ttab[fn_] := shrink[fn /@ Block[{n = Cases[fn, HoldPattern@Part[_, n_] :> n, 99] // Max}, PadLeft[#, n, False] & /@ (IntegerDigits[#, 2] & /@ Range[0, 2^n - 1]) /. {0 -> False, 1 -> True}]] In[382]:= shrink[L_List] := NestWhile[Transpose[Partition[#, 2]][[1]] &, L, Riffle[#2, #2] == #1 &, 2, \[Infinity], -1] Clear@cull; cull[L_List] := #[[1]] & /@ DeleteDuplicates[Prepend[ttab@#, #] & /@ L, Rest@#1 == Rest@#2 &] --rwg

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[math-fun] Draft of my January 2017 blog post
by James Propp 12 Jan '17

12 Jan '17

Hi, I started writing a new draft titled "Avoiding chazakah with the Prouhet-Thue-Morse sequence" and would love to get your feedback. I plan on publishing it on the 17th (sorry to be sending this out so late!). I would be especially interested in knowing about more good visuals (and "audibles"!) related to the Prouhet-Thue-Morse sequence. (I'm trying to get the publisher of Per Norgard's "Drumbook" to give me permission to use an excerpt from the accompanying CD, but I don't know if that will come through in time.) Suggestions for references, and comments of all kinds, are welcome. As usual, the piece is too long, so one useful form of feedback would be "The part about X isn't bad, but it's not as interesting as the other parts" (especially if many pre-readers single out the same X). And, more generally, I appreciate candid criticism from people who are sympathetic to my aims but think I've fallen short of them. My prose style is designed to make the material accessible, but my breeziness should not be interpreted as indicating indifference to scholarly correctness. Keep in mind that all math-fun feedback goes into one mail-feed, so I won't know whose feedback is whose unless you sign your comment. Also, all substantive suggestions that I use will be acknowledged (unless you specifically ask me not to do this). Please leave your feedback here: https://mathenchant.wordpress.com?p=1422&shareadraft=587802aa41b63 Title: Avoiding chazakah with the Prouhet-Thue-Morse sequence Beginning: The ingredients of today's midwinter mathematical stew will include beans, boats, never-ending chess games, a Dane named Per who's into aperiodic percussion, an ABBA that hails from Norway rather than Sweden, a song from Jewish summer camps, and a way to prove 0^2 + 3^2 + 5^2 + 6^2 = 1^2 + 2^2 + 4^2 + 7^2 using calculus instead of a calculator ... Read more: https://mathenchant.wordpress.com?p=1422&shareadraft= 587802aa41b63 Thanks, Jim Propp

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[math-fun] [EXTERNAL] terrestrial estimation question
by Bill Gosper 12 Jan '17

12 Jan '17

[Mathematica 11.1] control= Volume of ocean Quantity[1.332*10^9, ("Kilometers")^3] control= seawater density Quantity[Interval[{1020, 1035}], ("Kilograms")/("Meters")^3] %%*% Quantity[Interval[{1.35864*10^21, 1.37862*10^21}], "Kilograms"] control= seawater specific heat Quantity[Interval[{3.926, 4.022}], ("Joules")/("Grams" "KelvinsDifference")] %%*% Quantity[Interval[{5.33402064*10^24, 5.54480964*10^24}], ("Joules")/("KelvinsDifference")] In[104]:= UnitConvert[Quantity[1.332*10^9, ("Kilometers")^3], "mi"] Out[104]= Quantity[3.19563794427132*10^8, ("Miles")^3] control= mass of atmosphere Quantity[5.1441*10^18, "Kilograms"] But I couldn't get it to tell me the specific heat of air. --rwg Date: 2017-01-12 05:47 From: "Cordwell, William R" <wrcordw(a)sandia.gov> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >From NOAA: According to the U.S. Geological Survey, there are over 332,519,000 cubic miles of water on the planet. A cubic mile is the volume of a cube measuring one mile on each side. Of this vast volume of water, NOAA's National Geophysical Data Center estimates that 321,003,271 cubic miles is in the ocean. R_e ~ 6371 km. V_ocean ~ 1.33799*10^9 km^3. V / (4*pi*r^2) ~ 2.62318 km. (barring arithmetical errors). That is small enough so that correcting for the earth being curved will not substantially change the answer. The Challenger Deep is ~ 11 km deep. BC -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Marc LeBrun Sent: Wednesday, January 11, 2017 11:18 PM To: math-fun <math-fun(a)mailman.xmission.com> Subject: [EXTERNAL] [math-fun] terrestrial estimation question Were the Earth smoothed, preserving volume, how deep would the ocean be?

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[math-fun] terrestrial estimation question
by Marc LeBrun 12 Jan '17

12 Jan '17

Were the Earth smoothed, preserving volume, how deep would the ocean be?

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[math-fun] Recent multiply-perfect numbers?
by Allan Wechsler 11 Jan '17

11 Jan '17

Achim Flammenkamp's Multiply Perfect Numbers page hasn't been updated since the beginning of 2014. Certainly GIMPS has announced a few new Mersenne primes since then, Flammenkamp's database is missing at least the corresponding few classically perfect numbers. Has there been any other progress?

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[math-fun] Thue-Morse sequence
by Bill Gosper 11 Jan '17

11 Jan '17

Nice! Re #1, #2, #3: At the risk of restating the obvious: https://en.wikipedia.org/wiki/Firing_squad_synchronization_problem --rwg Date: 2017-01-10 17:30 From: Fred Lunnon <fred.lunnon(a)gmail.com> That crap Fig. 10 graphic is bugging me, so I have uploaded up a folder of colour graphics related to number walls to https://www.dropbox.com/sh/mkpkoqy8r6ybmpd/AAAaHPx8wnEdSvc3UMMp0uzCa including a cleaner view of the ternary Pagoda wall filed at pagodab.gif , colour-coded 0, 1, 2 -> blue, magenta, cyan. Note how zero/blue pixels are isolated, apart from the first two (constant) rows; a segment from further along the sequence appears on the third row. "Ternary" means computing the wall entries (Hankel determinants) modulo 3 ; so the walls over the integers also have only isolated zeros --- as also apparently do those modulo a lot of other primes congruent to -1 (mod 4). WFL On 1/10/17, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > Section 3 of the attached link might be relevant --- it turns out that > the > properties of T-M are much easier to establish via inspection of an > equivalent 4-symbol sequence: > > https://www.dropbox.com/s/1etnzbciqhbmy2j/csp08_submission_23.pdf > > And while I am on the subject, the main topic of that paper is a > little-known > analogue of T-M christened the "Pagoda" sequence, arising as follows. > The ternary T-M property of square-freedom is a special case of the more > restrictive property of minimal linear complexity, that is nowhere > satisfying > _any_ linear recurrence of order r (for more than 2r consecutive > terms). > > This constraint turns out impossible to satisfy for a ternary sequence; > but if relaxed to allow just 2r+1 consecutive terms (linear deficiency = > 2), > it results in an apparently unique (up to some fairly trivial > transformations) > object with a rather beautiful, fractal-like number wall. [Sadly the > publisher > required greyscale graphics, though these walls look much nicer in colour.] > > The paper finishes with an improbable 2-D tiling-based proof that > Pagoda satisfies deficiency = 2 , by which stage it has to be admitted > that the going has become a little heavy ... > > [Correction: the ambitious graphic on p.14 is missing its bottom > two rows, especially confusing since it was (deliberately) rotated > anticlockwise to emphasise the symmetry: as a result, the initial "22" > below has been omitted. Along with an initial invisible white (= 0) > and black (= 1) line running up the left-hand side from the bottom, > not to mention the eye-watering pixel scale, it was not a success. > > The Pagoda sequence should actually commence: > 22010211 22011201 22010211 02211201 > 22010211 22011201 02210211 02211201 ... ] > > Fred Lunnon > > > On 1/8/17, James Propp <jamespropp(a)gmail.com> wrote: >> Do any of you have any favorite private facts about the Thue-Morse >> sequence, or any favorite links to existing content on this subject? >> >> (I already know about the Numberphile video "The Fairest Sharing Sequence >> Ever".) >> >> I'd especially like a link to an image or animation graphically depicting >> the self-similarity of the Thue-Morse sequence. (I have my own ideas for >> a >> GIF that would depict this, but I prefer not to create things that >> already >> exist.) >> >> I'd also be interested in knowing whether any well-known (or not so >> well-known) poems use an abbabaab rhyme scheme, or whether there is any >> interesting music based on the Thue-Morse sequence. >> >> (Yes, this is all for my next Mathematical Enchantments piece.) >> >> Thanks, >> >> Jim Propp >

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[math-fun] Mallarme and Thue-Morse
by James Propp 11 Jan '17

11 Jan '17

Thanks, everyone! I now have all the leads I need. The link to the Oulipo website was especially helpful. (I meant to send this email to the whole list yesterday evening, but it turns out that I accidentally sent it only to Eric, with the result that Dan Asimov and Jean-Paul Delahaye and others thought I was still seeking help; apologies for that.) Jim ---------- Forwarded message ---------- From: *Eric Angelini* <Eric.Angelini(a)kntv.be <javascript:_e(%7B%7D,'cvml','Eric.Angelini(a)kntv.be');>> Date: Tuesday, January 10, 2017 Subject: Mallarme and Thue-Morse To: "jamespropp(a)gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp(a)gmail.com');>" < jamespropp(a)gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp(a)gmail.com');>> Hello James, here you have 7 pages (!) of such ABBA BAAB rhyming schema: http://blogs.oulipo.net/qc/category/formule-de-rimes/quatrain/q16-abba-baab/ I guess the most famous author here is Paul Valery: Je ne veux pas que tu sois triste (title = "I don't want you to be sad") Je ne veux pas que tu sois triste Puisque tu sais que je le suis, Ma lointaine, moi qui ne puis Ignorer que l’amour existe Depuis que mes jours et mes nuits De tes yeux mirent l’améthyste Et m’agitent cette batiste Qui voilait ce que je poursuis … Amèrement ma bouche avide Baise une place que je vois : Mes mains se perdent dans le vide Où se perd l’ombre de ma voix … Ö fantôme du château mien, Ne sois pas triste : c’est mon bien. Best, É. (going now to bed) ________________________________________ De : math-fun [math-fun-bounces(a)mailman.xmission.com] de la part de James Propp [jamespropp(a)gmail.com] Date d'envoi : mardi 10 janvier 2017 23:24 À : math-fun Objet : [math-fun] Mallarme and Thue-Morse Do any of you know any comp lit or poetry afficionados who'd have access to the first draft of Mallarme's poem "Le Pitre Chatie", which I gather used an abbabaab rhyme scheme for its first eight lines? I would like to use it in my essay on the Thue-Morse sequence. As always, I prefer using a couple of minutes of someone else's time (and giving them credit) instead of spending an hour of my own time (especially since a book giving preliminary versions of Mallarme's poems might be in French and hence hard for me to navigate). Thanks, Jim Propp _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Mallarme and Thue-Morse
by James Propp 10 Jan '17

10 Jan '17

Do any of you know any comp lit or poetry afficionados who'd have access to the first draft of Mallarme's poem "Le Pitre Chatie", which I gather used an abbabaab rhyme scheme for its first eight lines? I would like to use it in my essay on the Thue-Morse sequence. As always, I prefer using a couple of minutes of someone else's time (and giving them credit) instead of spending an hour of my own time (especially since a book giving preliminary versions of Mallarme's poems might be in French and hence hard for me to navigate). Thanks, Jim Propp

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[math-fun] Thue-Morse sequence
by James Propp 10 Jan '17

10 Jan '17

Do any of you have any favorite private facts about the Thue-Morse sequence, or any favorite links to existing content on this subject? (I already know about the Numberphile video "The Fairest Sharing Sequence Ever".) I'd especially like a link to an image or animation graphically depicting the self-similarity of the Thue-Morse sequence. (I have my own ideas for a GIF that would depict this, but I prefer not to create things that already exist.) I'd also be interested in knowing whether any well-known (or not so well-known) poems use an abbabaab rhyme scheme, or whether there is any interesting music based on the Thue-Morse sequence. (Yes, this is all for my next Mathematical Enchantments piece.) Thanks, Jim Propp

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[math-fun] Detecting binomial coefficients
by Dan Asimov 10 Jan '17

10 Jan '17

Suppose we are given the prime factorization of a rather large integer Q. Is there a good algorithm for determining from this whether the number Q is a binomial coefficient? I.e., whether there exist positive integers k < n such that Q = n! / (k! (n-k)!) . —Dan

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