This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1915 Excerpt: ...that p--r has no longer (the number of different letters exceeding 2) any simple value for different classes of permutations. It is not possible to obtain expressions for %afl and %xp+r in the foregoing manner. 110. The Parity of the Greater Index. The permutations of a" '7... are such that the number with an even ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1915 Excerpt: ...that p--r has no longer (the number of different letters exceeding 2) any simple value for different classes of permutations. It is not possible to obtain expressions for %afl and %xp+r in the foregoing manner. 110. The Parity of the Greater Index. The permutations of a" '7... are such that the number with an even index is equal to the number with an uneven index whenever more than one of the numbers k, ... are uneven. Consider the function (l)(2)...(i+j+k +...) (l)(2)...(i)(l)(2)...(j)(l)(2)...(k)...' if we put, therein, x =--1 and the function happens to vanish it must be because the permutations with even index are equal in number to those with uneven index. This is the case because the function is equal to %Gpxp where there are Cp permutations having the index equal to p. Under what conditions does x =--1 cause the function to vanish? The factor (m) becomes 2 for x =--1 whenever m is uneven. Eliminating these factors we are left with a function which may be of the form when x =--1 and can be evaluated. This happens when numerator and denominator contain the same number of factors (to), wherein To is even. If this be not the case the numerator must contain more of such factors than the denominator and the function must vanish for x=--1. Observe that the numerator cannot involve fewer of such factors than the denominator, for in that case the function would become infinite for x =--1, which from the nature of the function is absurd. We have, then, to find out the circumstances under which the numerator of the complete function involves more factors (to) than the denominator, To being even. If we consider the product (1)(2)...(-) and denote by Ea the greatest integer in a, it will be seen that the number of factors (to), To even, in the prod...
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