From Permutation Patterns to the Periodic Table Lara Pudwell Permutation patterns is a burgeoning area of research with pattern packing results in the general case before we con- roots in enumerative combinatorics and theoretical com- sider packing in a specific type of permutation that leads to puter science. Although much current work in this area a new and surprising connection with physical chemistry. still relates to its computer science origins, its rapid expan- Let 풮푘 be the set of all permutations on [푘] = {1, 2, … , 푘}. sion over the past two decades has led to some surprising Given 휋 ∈ 풮푘 and 휌 ∈ 풮ℓ we say that 휋 contains 휌 as a connections with other areas of mathematics. This article pattern if there exist 1 ≤ 푖1 < 푖2 < ⋯ < 푖ℓ ≤ 푘 such that first presents a brief overview of pattern avoidance anda 휋푖푎 ≤ 휋푖푏 if and only if 휌푎 ≤ 휌푏. In this case we say that survey of enumeration results that are standard knowledge 휋푖1 ⋯ 휋푖ℓ is order-isomorphic to 휌, and that 휋푖1 ⋯ 휋푖ℓ is an within the field. Then, we turn our attention to a newer occurrence or a copy of 휌 in 휋. If 휋 does not contain 휌, then optimization problem of pattern packing. We survey we say that 휋 avoids 휌. For example 휋 = 43512 contains the pattern 휌 = 231 because the digits of 휋2휋3휋5 = 352 have Lara Pudwell is an associate professor of mathematics and statistics at Valparaiso the same relative order as the digits of 휌; this is one of four University. Her email address is [email protected]. instances of 231 in 휋. Communicated by Notices Associate Editor Emilie Purvine. The definition of pattern containment may be made For permission to reprint this article, please contact: more visual by considering the plot of 휋. In particular, for [email protected]. 휋 = 휋1휋2 ⋯ 휋푘 ∈ 풮푘, the plot of 휋 is the graph of the points DOI: https://doi.org/10.1090/noti2115 994 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 (푖, 휋푖) in the Cartesian plane. The plots of 휋 = 43512 and The fact that s푘(12) = s푘(21) is a special case of a more ∗ 휌 = 231 are given in Figure 1. An alternate way to see that general phenomenon. If s푘(휌) = s푘(휌 ) for all 푘, we 휋 contains 휌 is to notice that if we remove the rows and say that 휌 and 휌∗ are Wilf-equivalent. A number of Wilf- columns with red points from the plot of 휋, the remaining equivalences are made clear by considering the plot of a (black) points form a plot of 휌. permutation as in Figures 1 and 2. All points in the plot of 휋 lie in the square [1, 푘] × [1, 푘], and thus we may ap- ply various symmetries of the square to obtain involutions 푟 on the set 풮푘. For 휋 ∈ 풮푘, we define 휋 = 휋푘 ⋯ 휋1 and 푐 휋 = (푘 + 1 − 휋1) ⋯ (푘 + 1 − 휋푘), which are called the reverse and complement of 휋, respectively. Notice that the plot of 휋푟 is obtained by reflecting the plot of 휋 across 푘+1 the vertical line at and 휋푐 is obtained by reflecting 2 푘+1 the plot of 휋 across the horizontal line at . We obtain 2 Figure 1. The plots of 휋 = 43512 and 휌 = 231. one additional important equivalence by considering 휋−1 which reflects the plot of 휋 over the line 푦 = 푥. For ex- 푟 푐 Of particular interest are the sets 풮푘(휌) = {휋 ∈ 풮푘 ∣ ample, the graphs of 휋 = 1342, 휋 = 2431, 휋 = 4213, 휋 avoids 휌}. For example, and 휋−1 = 1423 are shown in Figure 2. Notice that if 휋 contains 휌, then 휋푟 contains 휌푟, 휋푐 contains 휌푐, and 휋−1 풮4(123) = {1432, 2143, 2413, 2431, 3142, 3214, 3241, −1 contains 휌 . Thus, these involutions on 풮푘 also provide 푟 푐 −1 3412, 3421, 4132, 4213, 4231, 4312, 4321}, bijections between 풮푘(휌), 풮푘(휌 ), 풮푘(휌 ), and 풮푘(휌 ). As a consequence, for any pattern 휌, 휌 is Wilf-equivalent to 휌푟, and 휋 = 43512 ∈ 풮5(123) since there is no increasing sub- 휌푐, and 휌−1. These Wilf-equivalences that follow from the sequence of length 3 in 휋. action of the dihedral group on the plot of 휋 are known as One early occurrence of pattern avoidance is the Erd˝os– trivial Wilf-equivalences. Szekeres theorem [9], which can be rephrased as follows: Let 퐼푎 = 12 ⋯ 푎 be the increasing permutation of length 푎 and let 퐽푏 = 푏(푏 − 1) ⋯ 1 be the decreasing permutation of length 푏. Then 풮푘(퐼푎) ∩ 풮푘(퐽푏) = ∅ if 푘 > (푎 − 1)(푏 − 1). The study of pattern-avoiding permutations in their own right was instigated by Knuth’s work in The Art of Computer Pro- 휋 = 1342 휋푟 = 2431 휋푐 = 4213 휋−1 = 1423 gramming [12] when he showed that a permutation 휋 ∈ 풮푘 is sortable after one pass through a stack if and only if Figure 2. The plots of 휋 = 1342, 휋푟 = 2431, 휋푐 = 4213, and 휋−1 = 1423. 휋 ∈ 풮푘(231). Pattern avoidance has also proven to be a useful language to describe inputs sortable through a va- riety of other machines. Generalizations of pattern avoid- By trivial Wilf-equivalence, we have that s푘(132) = ance have been used to characterize geometric properties s푘(213) = s푘(231) = s푘(312) and that s푘(123) = s푘(321). of Schubert varieties. More recently, researchers have con- (2푘) 푘 In fact, it turns out that s푘(휌) = for 휌 ∈ 풮3. Consider nected permutation patterns with results in genomics, sta- 푘+1 tistical mechanics, and more. the case of s푘(132). We have s0(132) = s1(132) = 1. Now, consider 휋 ∈ 풮푘(132) and suppose that 휋푖 = 푘. For any Counting 휋푎 and 휋푏 with 푎 < 푖 < 푏, it must be that 휋푎 > 휋푏; oth- Much of the existing literature in permutation patterns erwise 휋푎푘휋푏 would form a 132 pattern. Since 휋1 ⋯ 휋푖−1 studies the quantity s푘(휌) = |풮푘(휌)| for various patterns and 휋푖+1 ⋯ 휋푘 must also avoid 132, this implies 휌. While the results in this section are central ideas in per- mutation patterns research, they are not closely related to 푘 the final result of this paper. The reader interested inacon- s푘(132) = ∑ s푖−1(132) ⋅ s푘−푖(132). 푖=1 nection to chemistry may wish to skip ahead to the section on pattern packing. (2푘) 푘 It is easily checked that s푘(휌) = satisfies this recur- Starting with the simplest case, it is immediate that 푘+1 s푘(1) = 0 if 푘 ≥ 1 since each digit of a nonempty per- rence and matches the required initial values. Notice that mutation is a copy of the pattern 1. We also have that s푘(휌) is the 푘th Catalan number. Pattern-avoiding permu- s푘(12) = s푘(21) = 1 for 푘 ≥ 0, since the unique permu- tations are just one of many enumerative combinatorics tation of length 푘 avoiding 12 (resp., 21) is 퐽푘 (resp., 퐼푘). contexts where the Catalan numbers appear; Stanley [16] AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 995 has collected more than 200 occurrences of this celebrated Definition. The packing density of 휌, denoted 푑(휌), is given sequence. There are also a number of bijections between by 풮 (123) and 풮 (132) that preserve various statistics; see, for max 휈(휌, 휋) 푘 푘 푑(휌) = lim 휋∈풮푘 . example, B´ona’s text [7]. 푘→∞ ( 푘 ) |휌| Although s푘(휌) depends only on 푘 and |휌| for patterns of length at most 3, the general case is more complicated. In the rest of this section, we survey the packing litera- It is known that if 휌 ∈ 풮4, then 휌 is Wilf-equivalent to ture. However, for our final connection to physical chem- 1342, 1234, or 1324, but s푘(1342), s푘(1234), and s푘(1324) istry, we care about packing into a specific kind of permu- are three distinct values for sufficiently large 푘. B´ona tation that is addressed in the next section. The reader in- [6] showed the sequence s푘(1342) has a nonrational al- terested more in this chemistry connection than in pattern gebraic generating function. Gessel [10] showed the se- packing in general may read the definition of sum below quence s푘(1234) has a nonalgebraic holonomic generat- and then skip ahead to the section on alternating permu- ing function. However, there is no known general for- tations. mula for s푘(1324), and better understanding the structure The most straightforward pattern of length 푎 to pack is of 풮 (1324) for large 푘 is an ongoing active area of research. 푘 the monotone increasing pattern. We have that 푑(퐼푎) = 1 The patterns we have discussed thus far are known since every subsequence of 퐼푘 of length 푎 is a copy of 퐼푎. For as classical permutation patterns. A number of varia- more general patterns, it may not initially be clear that 푑(휌) tions have been considered, including consecutive pat- exists, but an unpublished argument of Fred Galvin (de- terns (where digits of 휌 must appear in adjacent positions max휋∈풮 휈(휌,휋) scribed by Price in [15]) shows that 푘 is non- of 휋), vincular patterns (a hybrid between classical and ( 푘 ) |휌| consecutive patterns), and bivincular patterns (which also increasing for 푘 ≥ |휌|. Since this sequence is also clearly place restrictions on the relative sizes of the digits form- bounded below by 0, it converges. ing a copy of 휌, not just on their positions).