Lucas aggregate supply function

In mathematics (and particularly in combinatorics), the major index of a permutation is defined as the sum of the positions of the descents of the permutation. In symbols, the major index of the permutation w is

$\mathrm{maj}(w)=\sum _{w(i)>w(i+1)}i.$

For example, if w is given in 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 statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of 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.) 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.

$\overline{)\begin{array}{cccccccc}& 0& 1& 2& 3& 4& 5& 6\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 1& 0& 1& 1& 1& 0& 0& 0\\ 2& 0& 1& 2& 1& 1& 0& 0\\ 3& 0& 1& 1& 2& 1& 1& 0\\ 4& 0& 0& 1& 1& 2& 1& 0\\ 5& 0& 0& 0& 1& 1& 1& 0\\ 6& 0& 0& 0& 0& 0& 0& 1\end{array}}$

References

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