On unfair permutations

Descent-inversion statistics in riffle shuffles

“…For example, letting x = (3, 1, 7, 4, 2, 6, 5), one can easily see that (3,1,7,2,6,5) is an alternating subsequence and that LA 7 (x) = 6. See [19] for a survey on the longest alternating subsequence problem.…”

“…Finally, the variance is found to be V ar(inv(X) = E[(inv(X)) 2 ] − (E[inv(X]) 2 , where the moments are as given in (3.6) and (3.12).…”

“…See [10] for a proof that both the number of descents and the number of inversions are asymptotically normal in a uniform setting and [16] for an extension of the results of [10] to generalized descents. Also, let us note that the number of inversions is also studied in random words [4,14] and in unfair permutations [2].…”

“…, X n ) = (1, 1, 2, 1, 2, 2, 1) be a sample from the uniform distribution over {1, 2} 7 . Then sorting according to digits preserving relative order for cards with the same digit gives the new configuration of cards as (1,2,4,7,3,5,6). In the usual permutation notation, the resulting permutation after the inverse shuffle is ) .…”