Lucas aggregate supply function: Difference between revisions

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In [[mathematics]] (and particularly in [[combinatorics]]), the '''major index''' of a [[permutation]] is defined as the sum of the positions of the [[Permutation#Ascents, descents and runs|descents]] of the permutation. In symbols, the major index of the permutation ''w'' is
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: <math>
\operatorname{maj}(w) = \sum_{w(i)>w(i+1)} i.
</math>
 
For example, if ''w'' is given in [[Permutation#Notation|one-line notation]] by ''w'' = 351624 (that is, ''w'' is the permutation of {1, 2, 3, 4, 5, 6} such that ''w''(1) = 3, ''w''(2) = 5, etc.) then ''w'' has descents at positions 2 (from 5 to 1) and 4 (from 6 to 2) and so maj(''w'') = 2 + 4 = 6.
 
This [[permutation statistic|statistic]] is named after [[Percy Alexander MacMahon|Major Percy Alexander MacMahon]] who showed in [[#CITEREFMacMahon1913|1913]] that the distribution of the major index on all permutations of a fixed length is the same as the distribution of [[Permutation#Inversions|inversions]]. That is, the number of permutations of length ''n'' with ''k'' inversions is the same as the number of permutations of length ''n'' with major index equal to ''k''. (These numbers are known as ''Mahonian numbers'', also in honor of MacMahon.<ref>M. Bóna, Combinatorics of Permutations, 2004, p. 43ff, ISBN 1-58488-434-7.</ref>)  In fact, a stronger result is true: the number of permutations of length ''n'' with major index ''k'' and ''i'' inversions is the same as the number of permutations of length ''n'' with major index ''i'' and ''k'' inversions, that is, the two statistics are equidistributed.  For example, the number of permutations of length 4 with given major index and number of inversions is given in the table below.
:<math>\begin{array}{c|ccccccc}
& 0&1&2&3&4&5&6 \\
\hline
0&1&0&0&0&0&0&0 \\
1&0&1&1&1&0&0&0 \\
2&0&1&2&1&1&0&0 \\
3&0&1&1&2&1&1&0 \\
4&0&0&1&1&2&1&0 \\
5&0&0&0&1&1&1&0 \\
6&0&0&0&0&0&0&1
\end{array}
</math>
== References ==
<references />
*{{Cite journal
| last1=MacMahon | first1=P. A. | author1-link=Percy Alexander MacMahon
| title=The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects
| year=1913
| journal=Amer. J. Math.
| volume=35
| number=3
| pages=281&ndash;322
| jstor=2370312}}.
 
{{DEFAULTSORT:Major Index}}
[[Category:Permutations]]
 
{{combin-stub}}

Latest revision as of 18:36, 28 July 2014



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I used to cart my leatherman around all the time and ultimately stopped when I realized most of the time I necessary to reduce something and I frequently made use of a modest pocket knife for that. I have owned each Leatherman tools and Gerber tools. The Gerber ones are DEFINATELY inferior to the Leatherman ones. Gerber doesn't make pretty fantastic knives in basic. They use soft, inferior grade steels. I picked up a Leatherman Core yesterday and am happy with it. I bought it particularly for the reason that it did NOT have a pair of scissors which I see as a waste of space and weight. I also like that the tools and blades lock open. a two.9-inch 420HC knife (HC stands for “high carbon,” which implies the knife will hold an edge superior) Keychain carry of the tiny and burly Leatherman PS4.

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