TY - JOUR

T1 - Lengths of monotone subsequences in a Mallows permutation

AU - Bhatnagar, Nayantara

AU - Peled, Ron

N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation π∈Sn is proportional to qInv(π) where q is a real parameter and Inv(π) is the number of inversions in π. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that q<1, our results apply when q is a function of n satisfying n(1-q)→∞. The case that n(1-q)=O(1) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order n√1-q, whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q. We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order 1/(1-q). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.

AB - We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation π∈Sn is proportional to qInv(π) where q is a real parameter and Inv(π) is the number of inversions in π. The case q=1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that q<1, our results apply when q is a function of n satisfying n(1-q)→∞. The case that n(1-q)=O(1) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order n√1-q, whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q. We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order 1/(1-q). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.

KW - 05A05

KW - 60B15

KW - 60C05

KW - 60F10

KW - 60F25

UR - http://www.scopus.com/inward/record.url?scp=84897363681&partnerID=8YFLogxK

U2 - 10.1007/s00440-014-0559-7

DO - 10.1007/s00440-014-0559-7

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AN - SCOPUS:84897363681

SN - 0178-8051

VL - 161

SP - 719

EP - 780

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -