;ò L‘Fc@sYdZdgd„Zdfd„ƒYZeƒZed„Zed„Zdd„ZdS( sDCombinatorics. $Id: Pascal.py,v 1.1 2007/03/08 23:49:49 eddy Exp $ icCsñ|djo td‚ny ||SWntj onX|dt|ƒf\}}xŽ||jo€y||}Wn#tj ot|ƒ|}nXd|}t|ƒdjo+|i |ƒ|t|ƒjpt ‚q[q[W|SdS(s•Returns the factorial of any natural number. Equivalent to reduce(lambda a,b:a*(1+b), range(num), 1L), except that it doesn't return a long unless it has to (or num is one) and it degrades gracefully when given invalid input. For agrguments < 0 this raises a ValueError. For other arguments < 1, you'll get the answer 1, as this is correct if your argument is valid and resolves the problem of what to do with invalid input. For other non-integer non-negative values, you'll get a computed positive value which probably isn't far off the Gamma function's value at one more than that non-integer. For valid input, i.e. non-negative integers, factorial(num) = Gamma(1+num), wherein Gamma(1+a) = integral(: exp(-t).power(a, t) ←t :{positive reals}) for arbitrary a in the complex plane, with a pole (of order 1) at each negative integer and a real humdinger of a singularity at infinity. isI only do naturalsiÿÿÿÿii N( snums ValueErrorscaches IndexErrorslensresultsis OverflowErrorslongsappendsAssertionError(snumscachesisresult((s(/home/eddy/.sys/py/study/maths/Pascal.pys factorials&     #sPascalcBs>tZdZhZd„Zd„Zd„ZeZd„ZRS(s]A dictionary describing Pascal's triangle. Keys are pairs of naturals; value for (n,m) is (n+m)! / n! / m! Given the symmetry, no value is actually stored for key (n,m) when m > n. Given the nature of Pascal's triangle, computing the value for a given (n,m) involves computing the values for all (i,j) with i<=n and j<=m. Representation depicts Pascal's triangle with one side as the top line, the other as the left column; a `row' of Pascal's triangle is thus a diagonal stripe. Only values we've actually needed are shown, so there's nothing above the diagonal; except on the top line, where a 1 is displayed above the last digit of the diagonal entry in each column. Note that neither this class nor its one instance is any part of this module's export: it is over-ridden by a curried form of chose(), below. cCs¼|\}}y|i||ƒ}Wntj oƒ}|GH|iddjo‚nt ||ƒt |ƒt |ƒ}||jo||f\}}n||i ||fs != s N( sresultsselfs_Pascal__valuessitemssnsmsvals factorialschecksappends joinfields(sselfsvalsmsnsresultscheck((s(/home/eddy/.sys/py/study/maths/Pascal.pyscheck~s$ -( s__name__s __module__s__doc__s_Pascal__valuess __getitem__s_Pascal__lookups__str__s__repr__scheck(((s(/home/eddy/.sys/py/study/maths/Pascal.pysPascal*s   #cCs||||fSdS(s²chose(N,i) -> N! / (N-i)! / i! This is the number of ways of chosing i items from among N, ignoring order of choice. Computed using a cached form of Pascal's triangle. N(srackstotalspart(stotalspartsrack((s(/home/eddy/.sys/py/study/maths/Pascal.pyschose‹scCs|iƒdS(N(srackscheck(srack((s(/home/eddy/.sys/py/study/maths/Pascal.pyscheck“scCs*tt||d„td|ƒƒƒSdS(sfA row of Pascal's triangle, optionally scaled, as a tuple. Required argument, tot, is the row index: Pascal(1+i)[1+j] = Pascal(i)[j] + Pascal(i)[1+j] give-or-take missing entries being presumed zero, with Pascal[0] = (1,). Optional second argument is an over-all scaling to apply to all entries in the row; thus sum(Pascal(n, .5**n)) == 1. cCst||ƒ|S(N(schosestsiss(sistss((s(/home/eddy/.sys/py/study/maths/Pascal.pyssiN(stuplesmapstotsscalesrange(stotsscale((s(/home/eddy/.sys/py/study/maths/Pascal.pysPascal•sN(s__doc__s factorialsPascalschosescheck(s factorialsPascalschosescheck((s(/home/eddy/.sys/py/study/maths/Pascal.pys?s $_