Ordered partitions avoiding a permutation pattern of length 3

Abstract

An ordered partition of [n] = {1, 2, . . . , n} is a partition whose blocks are endowed with a linear order. Let OP n,k be set of ordered partitions of [n] with k blocks and OP n,k (σ) be set of ordered partitions in OP n,k that avoid a pattern σ. Recently, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [n] with 3 blocks and the number of ordered partitions of [n] with n − 1 blocks avoiding a permutation pattern of length 3. They showed that |OP n,k (σ)| = |OP n,k (123)| for any permutation σ of length 3, and raised the question concerning the enumeration of OP n,k (123). They also conjectured that the number of ordered partitions of [2n] with blocks of size 2 avoiding a permutation pattern of length 3 satisfied a second order linear recurrence relation.In answer to the question of Godbole, et al., we obtain the generating function for |OP n,k (123)| and we prove the conjecture on the recurrence relation.

Patterns in words of ordered set partitions

“…For example, they showed that op n,k (σ) = op n,k (123) for all permutations σ of length 3. They also proved that Later, Chen, Dai and Zhou [2] proved that…”

“…Later, Chen, Dai and Zhou [2] proved that 1 + n≥1 t n n k=1 op n,k (123)x k = −x + 2xt − 2t + 2t 2 x + 2t 2 + x √ 1 − 4xt − 4t + 4t 2 x + 4t 2 2t(x + 1) 2 (t − 1)…”

The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2, . . . , n Are Always Algebraic

“…The following argument is inspired by the beautiful proof in [3], but is phrased in such a way that will make it transparent how to generalize it for general r. Let g(x) be the weight-enumerator of W 2 . Recall that W 2 is the set of all 231-avoiding words whose letters consist of {1, 1, .…”

“…By general 'holonomic nonsense' ( [13]) it is known beforehand that there is some such linear recurrence, and it is possible to bound the order, thereby justifying, a posteriori, the guessed recurrence, provided that it is checked for sufficiently many initial values. A more direct proof was given by Chen, Dai, and Zhou ( [3]), who proved the stronger statement that the generating function is algebraic, and even found the defining equation explicitly:…”

Pattern Avoidance in Ordered Set Partitions

“…, n} into k blocks, written OP n,k , which is the set of partitions p such that p ⊢ [n], ℓ(p) = k, and where the order of blocks is important. For example 13/2/4 and 4/13/2 are two distinct members of OP 4,3 . In the sequel, we also let OP n = ∪ Given a partition p ∈ OP n , and a permutation ρ ∈ S m , we say that p contains ρ if there exist blocks B i 1 , .…”