The Distribution of Double Deficiencies in
Pattern-Avoiding Permutations
By Tipaluck Krityakierne, Thotsaporn "Aek" Thanatipanonda, and Doron Zeilberger
.pdf
First Written: Sept. 10, 2026.
We study the distribution of the number of double deficiencies (DD) in permutations avoiding one or two patterns of length 3. Using structural decompositions of
these avoidance classes-together with a lattice-path decomposition in the 321-avoiding
case-we derive functional equations and convolution-type recurrences that efficiently
compute the corresponding double-deficiency generating functions in all but one singlepattern case. In the 321-avoiding case, we obtain an algebraic generating function,
from which we derive exact formulas for the mean and variance and prove that the
distribution is asymptotically binomial. We also identify a DD-preserving symmetry
that yields DD-Wilf equivalences, reducing the number of two-pattern cases that need
to be considered separately. For the resulting two-pattern classes, we obtain explicit
recurrences, including C-finite relations.
Added Sept. 11, 2026: Read
Per Alexandersson's insightful remarks,
Maple package
Sample Input and Output Files for DD3Avoid.txt