The set (§26.13) can be viewed as the collection of all ordered lists of elements of : . As an example, 35247816 is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
Equivalently, this is the sum over of the number of integers less than that lie in positions to the right of the th position:
A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. The permutation 35247816 has two descents: 52 and 81. The major index is the sum of all positions that mark the first element of a descent:
For example, . The major index is also called the greater index of the permutation.
The Eulerian number, denoted , is the number of permutations in with exactly descents. An excedance in is a position for which . A weak excedance is a position for which . The Eulerian number is equal to the number of permutations in with exactly excedances. It is also equal to the number of permutations in with exactly weak excedances. See Table 26.14.1.
|10||1||1013||47840||4 55192||13 10354||13 10354||4 55192||47840||1013||1|