Major index
Encyclopedia
In mathematics
, the major index of the permutation
π is defined as
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the major index of the permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
π is defined as
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In other words, the major index of π is the sum of the positions of its “descent tops.” For example, the major index of the permutation 25143 (in one-line notation) is 2 + 4 = 6.
This statistic is named after Major Percy Alexander MacMahonPercy Alexander MacMahonPercy Alexander MacMahon was a mathematician, especially noted in connection with the partitions of numbers and enumerative combinatorics.-Early life:...
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.