Summary of results for qVATTER

The following data were generated by Andrew Baxter's q-interpretation of enumeration schemes. Given an enumeration scheme (as computed in Zeilberger's VATTER package) to compute the number of permutations avoiding a set of patterns B, Baxter's qVATTER package can compute the generating function where the coefficient of qk is the number of permutations of length n avoiding B with k inversions. Each output page linked below was computed by the qSipur procedure in the qVATTER package. This computed the equivalence classes of patterns (or sets of patterns) of a given size (or set of sizes) under the trivial maps of taking reverses, complements, and inverses. Then for each class it tried to find a representative, B, which had a finite scheme, using procedures from VATTER. It then used this scheme to compute the following information:
  1. The scheme for avoiding B
  2. A sequence of generating functions, where the coefficient of qk of the nth polynomial is the number of permutations of length n avoiding B with k inversions. This is computed for 1 < n < 15.
  3. Since taking the reverse or complement changes the number of inversions (both in the same way), a second sequence of generating functions is given. These are for the set of patterns Br, where one takes the reverse of each pattern in B. These are computed by replacing q by 1/q in the first sequence, and then multiplying the nth term by qbinomial(n,2). This transformation reflects the change in the number of inversions.
  4. The specialization q=1, which is the same for both sequences of polynomials. This is simply the number of permtuations avoiding B, exactly what the original VATTER computes.
  5. By specializing q=1, q=-1, taking the sum, and dividing by 2 we can compute the number of even permutations of length n avoiding B.
  6. Similarly we can compute the number of odd permutations of length n avoiding B
  7. Due to the change in the number of reverses, the number of even/odd permutations avoiding Br is different from the number of those avoiding B. Thus these sequences are also given.
  8. By standard transformations, we can compute the statistical moments of these sequences. First we compute the average number of inversions for each length n.
  9. We next compute the standard deviation at each n.
  10. These allow us to compute the centralized moments. First, however, is the mislabeled variance.
  11. Skewness is the third centralized moment.
  12. Kurtosis is the fourth centralized moment.
Following the (attempted) computation for each equivalence class, a summary is presented of the success rate (duplciated in the table below) along with those classes which did not admit a finite scheme (deemed "failures"). These exactly match the rates from the package VATTER, but are repeated for completeness.


Pattern Set Structure Success Rate
One 3-pattern 2/2 (100%)
Pair of 3-patterns 5/5 (100%)
Trio of 3-patterns 5/5 (100%)
One 4-pattern 2/7 (28.6%)
Pair of 4-patterns 29/56 (51.8%)
Trio of 4-patterns 235/317 (74.1%)
(Mixing pattern lengths:)
A 3-pattern and a 4-pattern 17/18 (94.4%)
A 4-pattern and a pair of 3-patterns 23/23 (100%)
A 3-pattern and a 5-pattern 34/42 (81.0%)