From 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 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 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 . 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). Barred pat- The main other type of permutation pattern for which terns, where 휋 avoids 휌 unless 휌 is part of a larger specified we can say something about 푑(휌) is that of layered patterns. pattern in 휋, have proved useful in characterizing permu- tations sortable after repeated passes through a stack and Definition. The sum of permutations 훼 = 훼1 ⋯ 훼푎 and in describing geometric properties of Schubert varieties. 훽 = 훽1 ⋯ 훽푏, denoted 훼 ⊕ 훽, is given by Other types of enumeration questions are also of interest. 훼푖 1 ≤ 푖 ≤ 푎, Researchers have enumerated pattern-avoiding words, set 훼 ⊕ 훽 = { 푎 + 훽 푎 + 1 ≤ 푖 ≤ 푎 + 푏. partitions, integer partitions, lattice paths, and other com- 푖−푎 binatorial objects with appropriate definitions of patterns. Definition. A permutation is layered if it can be written in Pattern avoidance itself is really a special case of the more the form 퐽푎1 ⊕ 퐽푎2 ⊕ ⋯ ⊕ 퐽푎푝 for positive integers 푎1, … , 푎푝. general question of studying The permutation shown in Figure 3 is an example of a layered permutation since it is of the form 퐽1 ⊕ 퐽2 ⊕ 퐽2 ⊕ 풮푘,푐(휌) = {휋 ∈ 풮푘 ∣ 휋 contains 푐 copies of 휌}. 퐽2 ⊕ 퐽1. However, the simplest nontrivial layered permu- tation pattern is 132 = 퐽1 ⊕ 퐽2. Stromquist [17] showed max 휈(휌,휋) that if 휌 is layered, then 휋∈풮푘 is achieved by a Thus far, we have focused on the situation when 푐 = 0. For ( 푙 ) |휌| some specific patterns 휌, |풮푘,푐(휌)| is known for 푐 ≥ 1, but layered 휋. Stromquist and later Barton [4] showed that in general, the pattern containment question is hard. The 푑(132) = 2√3 − 3 ≈ 0.464 by focusing on layered 휋 ∈ 풮푘 literature on pattern-avoiding permutations is vast, but in max 휈(휌,휋) and showing that 휋∈풮푘 is achieved when the top the remainder of this paper we focus on pattern packing. ( 푘 ) |휌| In other words, given a pattern 휌, we consider the maxi- 3−√3 layer is of size 푎 ≈ ( ) 푘 and the remaining entries are mum value of 푐 (given as a function of 푘) for which 풮푘,푐(휌) 2 is nonempty. filled in recursively with an optimal 132-packing layered permutation of length 푘 − 푎. Packing For patterns of length 4, Price [15] showed that 푑(1432) In this section, we consider the maximum number of is given by the real root of 푥3 − 12푥2 + 156푥 − 64 and that 휌 푘 3 copies of that can occur in a permutation of length . 푑(2143) = . Later, Albert et al. [1] showed that 푑(1243) = 휈(휌, 휋) 휌 휋 8 Let be the number of occurrences of in . If we wish 3 max 휈(휌, 휋) . Bounds on 푑(1324), 푑(1342), and 푑(2413) are known to determine 휋∈풮푘 , it is advantageous to report 8 the percentage of copies of 휌 out of all subsequences of but have not yet been proved to be sharp. In the case of length |휌| in 휋 rather than just the number of copies. 1324, this illustrates that even if a pattern is layered, that

996 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 does not imply the packing density is straightforward to To see the connection with physical chemistry, the last determine. A variety of additional packing literature exists, problem we will consider is packing 퐼푎 into an alternating both packing patterns into permutations and into words permutation of length 푘. We already know the optimal (i.e., permutations with repeated digits) [5, 8, 11,18–20], way to pack 퐼푎 into an arbitrary permutation of length 푘 but in general packing problems have seen slower progress is to pack it into 퐼푘, where every subsequence of length 푎 than pattern avoidance problems. is a copy of 퐼푎. However, 퐼푘 ∉ 풜푘. To address this, we construct the alternating permutation 퐼ˆ푘 defined as Packing in Alternating Permutations Rather than focusing on packing in all permutations, in ⎧퐽1 ⊕ 퐽⏟⎵⎵⏟⎵⎵⏟2 ⊕ ⋯ ⊕ 퐽2 ⊕퐽1 푘 even, 푘 the rest of this paper we will focus on packing patterns ⎪ −1 times 2 into permutations with extra restrictions. This family of 퐼ˆ푘 = ⎨퐽1 ⊕ 퐽⏟⎵⎵⏟⎵⎵⏟2 ⊕ ⋯ ⊕ 퐽2 푘 odd. packing problems will provide a new link between permu- ⎪ 푘−1 times tations and physical chemistry. ⎩ 2 Definition. Permutation 휋 is an alternating permutation if As an example, the plot of 퐼ˆ8 is given in Figure 3. 휋1 < 휋2 > 휋3 < 휋4 ⋯ . Alternating permutations are also known as zig-zag per- mutations or up-down permutations. They were first stud- ied by Andr´e[2,3] in the nineteenth century. Let 풜푘 be the set of alternating permutations of length 푘. Andr´eproved the surprising result that ∞ |풜 | 푥푘 ˆ ∑ 푘 = sec(푥) + tan(푥), Figure 3. The plot of 퐼8. 푘! 푘=0 giving a natural combinatorial interpretation to a series ex- 휈(퐼푎, 퐼ˆ푘) Although < 1 for specific values of 푘, we claim pansion of trigonometric functions. (푘) Now, instead of the classical definition of packing den- 푎 that 퐼ˆ is the construction that proves 푑 (퐼 ) = 1. For the sity above, we consider the following modification. 푘 퐴 푎 case where we pack 퐼3 = 123 into 휋 = 퐼ˆ푘, notice that every Definition. The alternating packing density of 휌, denoted subsequence of length 3 still forms a 123 pattern unless 푑퐴(휌), is two digits of the subsequence come from the same layer max 휈(휌, 휋) 푘 휋∈풜푘 of 퐼ˆ푘. While there are ( ) total subsequences of length 3 푑퐴(휌) = lim . 3 푘→∞ ( 푘 ) 푘 | | in 퐼ˆ푘, there are less than 2( ) subsequences with two digits 휌 2 푘 푘 from the same layer of 퐼ˆ푘. Therefore 휈(퐼3, 퐼ˆ푘) ≥ ( ) − 2( ) Although it is not necessary for the upcoming chemistry 3 2 result, the interested reader may wish to compare 푑퐴(휌) to and in the limit, the classical packing density 푑(휌) introduced in the previ- 푘 푘 max휋∈풮 휈(휌,휋) ( ) − 2( ) 푘 3 2 푘 − 8 ous section. We know 푑(휌) exists because 푘 is ( ) 1 ≥ 푑퐴(123) ≥ lim 푘 = lim = 1. |휌| 푘→∞ ( ) 푘→∞ 푘 − 2 nonincreasing in 푘. However, this is no longer the case for 3 max휋∈풜 휈(휌,휋) 푘 when we restrict our attention to alternating Since a length 3 subsequence of 퐼ˆ fails to be a 123 pattern ( 푘 ) 푘 |휌| only when it uses adjacent digits that were required to be permutations. If 푑 (휌) exists, we immediately have that 퐴 in decreasing order by the definition of alternating permu- 푑 (휌) ≤ 푑(휌) for any pattern 휌 since 풜 ⊆ 풮 . Strikingly, 퐴 푘 푘 tation, 퐼ˆ is, in fact, the alternating permutation of length it turns out that 푑 (휌) = 푑(휌) for any pattern 휌. Since 푘 퐴 푘 with the maximum number of copies of 123. Finding 푑(휌) exists, we may construct a sequence of permutations the exact value of 휈(123, 퐼ˆ푘) is a straightforward exercise in 휈(휌, 휏푘) with 휏푘 ∈ 풮푘 such that lim = 푑(휌). The proof that cases. We have: 푘→∞ ( 푘 ) |휌| 푑퐴(휌) = 푑(휌) relies on a construction that replaces each Theorem. The maximum number of copies of 123 in an alter- 푘 point in the plot of 휏푘 with an alternating permutation to nating permutation of length is given by produce a sequence of alternating permutations 휎푘 with (푘−2)(푘2−4푘+6) 휈(휌, 휎 ) 푘 even, lim 푘 = 푑(휌) ˆ 6 푘 as well. 휈(123, 퐼푘) = { (푘−1)(푘−2)(푘−3) 푘→∞ ( ) 푘 odd. |휌| 6

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 997 1IA 18 VIIIA

1 2 H 2IIA 13 IIIA 14 IVA 15 VA 16 VIA 17 VIIA He (1,0) (1,0)

3 4 5 6 7 8 9 10 Li Be B C N O F Ne (2,0) (2,0) (2,1) (2,1) (2,1) (2,1) (2,1) (2,1)

11 12 13 14 15 16 17 18 Na Mg 3 IIIA 4 IVB 5 VB 6 VIB 7 VIIB 8 VIIIB 9 VIIIB 10 VIIIB 11 IB 12 IIB Al Si P S Cl Ar (3,0) (3,0) (3,1) (3,1) (3,1) (3,1) (3,1) (3,1) 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr (4,0) (4,0) (3,2) (3,2) (3,2) (3,2) (3,2) (3,2) (3,2) (3,2) (3,2) (3,2) (4,1) (4,1) (4,1) (4,1) (4,1) (4,1)

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe (5,0) (5,0) (4,2) (4,2) (4,2) (4,2) (4,2) (4,2) (4,2) (4,2) (4,2) (4,2) (5,1) (5,1) (5,1) (5,1) (5,1) (5,1) 55 56 57 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn (6,0) (6,0) (5,2) (5,2) (5,2) (5,2) (5,2) (5,2) (5,2) (5,2) (5,2) (5,2) (6,1) (6,1) (6,1) (6,1) (6,1) (6,1) 87 88 89 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg Cn Uut Fl Uup Lv Uus Uuo (7,0) (7,0) (6,2) (6,2) (6,2) (6,2) (6,2) (6,2) (6,2) (6,2) (6,2) (6,2) (7,1) (7,1) (7,1) (7,1) (7,1) (7,1)

58 59 60 61 62 63 64 65 66 67 68 69 70 71 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu atomic (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) (4,3) no. Symbol (n, ) 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3) (5,3)

Figure 4. The periodic table of chemical elements.

푘 Proof. In the case when 푘 is even, there are − 1 layers of While this argument is not complicated, the numbers 2 themselves are interesting. Table 1 gives the values of size 2 and two layers of size 1 in 퐼ˆ푘. A copy of 123 can be 휈(123, 퐼ˆ푘) for 4 ≤ 푘 ≤ 10. This is sequence A168380 formed by using both, one, or neither layer of size 1. If we in the On-Line Encyclopedia of Integer Sequences [14]. use both layers of size 1, it remains to choose a layer of size However, the primary description is perhaps surprising: 2 and then choose one point from the layer to complete this is the sequence of atomic numbers of the alkaline the 123 pattern. If we use one layer of size 1, we must first earth metals of the periodic table. For emphasis, see Fig- choose the layer of size 1, then choose two of the layers of ure 4. The values of 휈(123, 퐼ˆ) correspond to the atomic size 2, and choose one point from each of those layers. If 푘 numbers in the second column, labeled as Group “2 we use no layers of size 1, then we need to choose three - IIA”. In other words, the quasi-polynomial sequence distinct layers of size 2 and then choose one point from 2, 4, 12, 20, 38, 56, 88, … not only gives values of 휈(123, 퐼ˆ), it each of those layers. Therefore, there are 푘 also gives the atomic numbers of helium, beryllium, mag- 푘 푘 nesium, calcium, and more. This is unexpected! Although 푘 − 1 − 1 (푘 − 2)(푘2 − 4푘 + 6) 2 ( − 1) + 8( 2 ) + 8( 2 ) = permutation patterns appear in a wide variety of contexts, 2 2 3 6 a connection to physical chemistry is novel and quite dif- ferent from other more abstract appearances of patterns. copies of 123 in 퐼ˆ when 푘 is even. 푘 In the rest of this paper we give a bijective proof connect- The odd case is similar, with the adjustment that there 푘−1 ing our pattern packing problem to the language of physi- are now layers of size 2 and one layer of size 1 in 퐼ˆ푘. 2 cal chemistry. 푘−1 There are 4( 2 ) ways to construct a 123 pattern using the 2 푘 4 5 6 7 8 9 10 initial layer of size 1 and two layers of size 2. Otherwise, we must pick three distinct layers of size 2 and then choose 휈(123, 퐼ˆ푘) 2 4 12 20 38 56 88 one point from each of those layers. Therefore, there are Table 1. Values of 휈(123, 퐼ˆ푘) for 4 ≤ 푘 ≤ 10. 푘−1 푘−1 (푘 − 1)(푘 − 2)(푘 − 3) 4( 2 ) + 8( 2 ) = 2 3 6

□ copies of 123 in 퐼ˆ푘 when 푘 is odd.

998 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 Physical Chemistry Although there is only one way to orient an ℓ = 0 PhysicalIn the periodic Chemistry table, the atomic number of an element is (spherical)Although there orbital is aroundonly one an way atom’s to orient nucleus, an ℓ subshells = 0 ℓ ≥ 1 the number of protons in an atom of that element. While (spherical)with orbitalcan be around oriented an in multipleatom’s nucleus, ways. The subshells magnetic In the periodic table, the atomic number of an element is 푚 various ions exist, the atomic number also gives the num- withquantumℓ ≥ 1 can number be orienteddescribes in multiple an orbital’s ways. The orientation magnetic in the number of protons in an atom of that element. While 푛 ℓ 푚 ber of electrons in a neutral atom. In Schrödinger’s model, quantumspace. Givennumber 푚anddescribes, is an reported orbital’s as orientation an integer such in various ions exist, the atomic number also gives the num- −ℓ ≤ 푚 ≤ ℓ ℓ = 1 electrons move in space about the atom’s nucleus, and space.that Given 푛 and.ℓ For, 푚 example,is reported a polar as an ( integer) suchorbital ber of electrons in a neutral atom. In Schrödinger’s model, 푚 = −1 푚 = 0 Schrödinger’s wave equation is used to predict possible thathas−ℓ three ≤ 푚 possible ≤ ℓ. For orientations, example, a i.e., polar (ℓ =, 1) orbital, and electrons move in space about the atom’s nucleus, and 푚 = 1 electron orbitals, that is, the regions where electrons are has three, which possible could orientations, be thought i.e., of푚 as orienting= −1, 푚 =this 0, orbital and Schrödinger’s wave equation is used to predict possible shape along the 푥-axis, 푦-axis, or 푧-axis in space. electronmost likely orbitals, to be that found. is, the In three-dimensional regions where electrons space, arethree 푚 = 1, which could be thought of as orienting this orbital variables are required to describe the size, shape, and orien- shapeFinally, along the there푥-axis, is a fourth푦-axis, quantum or 푧-axis in number space. that is inde- most likely to be found. In three-dimensional space, three pendent of the other three called the spin quantum num- variablestation of are an required orbital. to These describe are the the principal size, shape, quantum and orien- num- Finally, there is a fourth quantum number that is inde- 푛 ℓ ber (푠). For any valid combination of (푛, ℓ, 푚), there are tationber( of), an the orbital. angular These quantum are the numberprincipal ( quantum), and the num- mag- pendent of the other three called the spin quantum num- 푚 two possible spins for an electron. bernetic (푛), quantum the angular number quantum ( ). number More details (ℓ), can and be the found mag- in ber (푠). For any valid combination of (푛, ℓ, 푚), there are any standard chemistry book such as [13]. two possibleIn summary, spins in for Schrödinger’s an electron. model of the atom, pairs netic quantum number (푚). More details can be found in (푛, ℓ, 푚) The principal quantum number 푛 corresponds to the ofIn electrons summary, are in in Schrödinger’s bijection with model tuples of of the integers atom, pairs any standard chemistry book such as [13]. 푛 ≥ 1 0 ≤ ℓ ≤ 푛 − 1 |푚| ≤ ℓ size of an electron orbital. 푛 is given as a positive integer, ofsuch electrons that are in bijection, with tuples, ofand integers (푛,. ℓ, How- 푚) The principal quantum number 푛 corresponds to the 푛 and larger values of 푛 correspond to larger orbitals. All suchever, that knowing푛 ≥ 1, the0 ≤ highest ℓ ≤ 푛-value − 1, and that|푚 appears| ≤ ℓ. in How- a par- size of an electron orbital. 푛 is given as a positive integer, ticular atom does not mean that all (푛, ℓ, 푚) tuples follow- andorbitals larger that values have of the푛 correspond same principal to largerquantum orbitals. number All are ever, knowing the highest 푛-value that appears in a par- said to be in the same shell of the atom. ticularing these atom inequalities does not mean appear. that Observe all (푛, ℓ, that 푚) tuples in the follow- periodic orbitals that have the same principal quantum number are ℓ The angular quantum number ℓ gives the shape of an ingtable these there inequalities are always appear. two shells Observe with that orbitals in the of periodic shape said to be in the same shell of the atom. before orbitals with shape ℓ + 1 are introduced. orbital.The angular While quantum multiple number electronsℓ maygives be the in the shape same of shell, an table there are always two shells with orbitals of shape ℓ the angular quantum number partitions the orbitals of a orbital. While multiple electrons may be in the same shell, beforeThe orbitals Bijection with shape ℓ + 1 are introduced. shell into subshells. Larger values of 푛 allow more room the angular quantum number partitions the orbitals of a ˆ for different shapes of subshells. Given 푛, ℓ is an integer TheConsider Bijection퐼푘 = 퐽1 ⊕ 퐽2 ⊕ ⋯. We wish to connect copies of shell into subshells. Larger values of 푛 allow more room ˆ 0 ≤ ℓ ≤ 푛 − 1 123 in 퐼ˆ푘 to valid tuples (푛, ℓ, 푚, 푠) of quantum numbers forwith different shapes of. subshells. The shapes Given of orbitals푛, ℓ is for an small integer val- Consider 퐼푘 = 퐽1 ⊕ 퐽2 ⊕ ⋯. We wish to connect copies of ues of ℓ are shown in Figure 5. To determine the (푛, ℓ) 123for in electrons퐼ˆ to valid in alkaline tuples (푛, earth ℓ, 푚, metals. 푠) of quantum numbers with 0 ≤ ℓ ≤ 푛 − 1. The shapes of orbitals for small val- 푘 푠 pairs that describe subshells in a particular atom, consider for electronsFirst, we in address alkaline the earth spin metals. number, . Just as there are ues of ℓ are shown in Figure 5. To determine the (푛, ℓ) (푛, ℓ, 푚) the coloring of the periodic table given in Figure 4. Each twoFirst, spin we numbers addressthe for any spin valid number, 푠. Justtuple, as observe there are that pairs that describe subshells in a particular atom, consider ˆ color corresponds to an ℓ value. Since ℓ ≤ 푛 − 1, the first twocopies spin numbers of 123 in for퐼푘 anycome valid in pairs.(푛, ℓ, 푚) Intuple, particular observe if 휋 that푎휋푏휋푐 the coloring of the periodic table given in Figure 4. Each 휋 = 퐼ˆ 휋 휋 휋 elements with a particular ℓ value have principal number copiesform of a 123in pattern퐼ˆ come in in pairs.푘, then In particular푎, 푏, and if 휋 휋푐 must휋 color corresponds to an ℓ value. Since ℓ ≤ 푛 − 1, the first 푘 휋 휋′ 푎 푏 푐 푛 = ℓ + 1. Subsequent rows of elements with a particu- formcome a 123 from pattern different in 휋 layers = 퐼ˆ of, then. Let휋 , 푏휋be, and the other휋 must digit elements with a particular ℓ value have principal number 휋 푘 휋 휋′ 휋푎 푏 푐 lar ℓ value increment 푛 by 1. This gives a straightforward comein the from same different layer as layers푏. Then of 휋. Let푎 푏휋′푐 beis a the second other 123 digit pat- 푛 = ℓ + 1. Subsequent rows of elements with a particu- ˆ 푏 ˆ mapping from an element to one (푛, ℓ) pair. Any (푛, ℓ) tern in 퐼푘. Two copies of 123 that′ use the same layer of 퐼푘 lar ℓ value increment 푛 by 1. This gives a straightforward in the same layer as 휋푏. Then 휋푎휋푏휋푐 is a second 123 pat- pair that corresponds to an element with a smaller atomic for theˆ second digit correspond to pairs of electrons thatˆ mapping from an element to one (푛, ℓ) pair. Any (푛, ℓ) tern in 퐼푘. Two copies of 123 that use the same layer of 퐼푘 have the same principal (푛), angular (ℓ), and magnetic (푚) pairnumber that corresponds is also possible to an in element the original with element. a smaller For atomic exam- for the second digit correspond to pairs of electrons that ℓ = 0 quantum numbers but different spin numbers. numberple, calcium is also (atomic possible number in the original 20) is shaded element. as For exam-. This have the same principal (푛), angular (ℓ), and magnetic (푚) ℓ = 0 Now, for the remaining quantum numbers, 푚 encodes ple,iscalcium the fourth (atomic row of numberorbitals. 20) is shaded Therefore, as ℓ calcium = 0. This has quantum numbers but different spin numbers. (푛, ℓ) = (4, 0) the value of the “1” in a 123 pattern, ℓ encodes the layer is theorbitals fourth with row of ℓ = 0 orbitals.. However, Therefore, looking calcium at the has ele- Now, for the remaining quantum numbers, 푚 encodes of the “2”, and 푛 encodes the value of the “3”. To be more orbitalsments with with(푛, smaller ℓ) =atomic (4, 0). However, numbers, looking we see calcium at the ele- also the value of the “1” in a 123 pattern, ℓ encodes the layer (푛, ℓ) ∈ {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)} precise, call the initial 퐽1 layer of 퐼ˆ푘 layer 0, and enumerate mentshas orbitals with smaller with atomic numbers, we see calcium also . of the “2”, and 푛 encodes the value of the “3”. To be more Notice that (푛, ℓ) = (3, 2) first occurs in scandium (atomic the remaining layers starting withˆ layer 1. A copy 휋푎휋푏휋푐 of has orbitals with (푛, ℓ) ∈ {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1)}. precise, call the initial 퐽1 layer of 퐼푘 layer 0, and enumerate number 21), so calcium has no (푛, ℓ) = (3, 2) subshell. 123 in 휋 = 퐼ˆ푘 maps to a tuple of integers in the following Notice that (푛, ℓ) = (3, 2) first occurs in scandium (atomic the remaining layers starting with layer 1. A copy 휋푎휋푏휋푐 of way: ˆ푚 is the layer of 퐼ˆ푘 where 휋푎 is found. Then number 21), so calcium has no (푛, ℓ) = (3, 2) subshell. 123 in 휋 = 퐼ˆ푘 maps to a tuple of integers in the following way: ˆ푚 is the layer− ˆ푚 of 휋퐼ˆ푘iswhere the smaller휋푎 is found. entry of Then layer ˆ푚, 푚 = { 푎 − ˆ푚ˆ푚 휋 휋is푎 theis the smaller larger entry entry of of layer layer ˆ푚,ˆ푚. 푚 = { 푎 ˆ푚 휋 ˆ푚. ℓ + 1 is the layer푎 ofis퐼ˆ푘 thewhere larger휋푏 entryis found. of layer Finally, given ℓ and 휋푐, we compute 푛 = 휋푐 −ℓ−3; intuitively, larger values ℓ + 1 is the layer of 퐼ˆ푘 where 휋푏 is found. Finally, given ℓ of 푛 correspond to a larger gap between the location of 휋푏 and 휋푐, we compute 푛 = 휋푐 −ℓ−3; intuitively, larger values and 휋푐 in 퐼ˆ푘, but with an appropriate offset to ensure 푛 is ℓ = 0 ℓ = 1 ℓ = 2 of 푛 correspond to a larger gap between the location of 휋푏 andpositive.휋 in 퐼ˆ, but with an appropriate offset to ensure 푛 is FigureFigure 5. 5.SimpleSimple electron electron orbital orbital shapes. shapes. 푐 푘 퐼ˆ = 퐽 ⊕ 퐽 ⊕ positive.As an example, consider copies of 123 in 7 1 2 퐽2 ⊕ 퐽2 = 1325476. There are 20 copies of 123 in 퐼ˆ7, and As an example, consider copies of 123 in 퐼ˆ7 = 퐽1 ⊕ 퐽2 ⊕ 퐽2 ⊕ 퐽2 = 1325476. There are 20 copies of 123 in 퐼ˆ7, and AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 999

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 999 these 20 copies correspond to the 20 electrons in an atom for 휈(123, 퐼ˆ푘) when 푘 > 10? What other chemical or physi- of calcium. Just as copies of 123 come in pairs, electrons cal structures can be described in terms of pattern packing in calcium come in 10 pairs that correspond to 10 differ- or pattern avoidance? Are there other combinatorial struc- ent tuples of quantum numbers (푛, ℓ, 푚). The 20 copies tures that give alternate ways to generate the sequences ˆ of 123 in 퐼7 are given in pairs with their corresponding of atomic numbers of particular groups of chemical ele- (푛, ℓ, 푚) tuples in Table 2. For instance, both 246 and 256 ments? The variety of applications of permutation patterns are copies of 123 in 퐼ˆ7. Both have their initial element in has grown tremendously in recent decades, and modeling layer 1, so both correspond to ˆ푚= 1. Since 2 is the smaller electron orbitals can now be added to the list. digit in this layer, we have 푚 = −1. We see that in both References copies 휋푗 comes from layer 2, so ℓ + 1 = 2, i.e., ℓ = 1. Fi- nally, 푛 = 6 − 1 − 3 = 2. So these two copies correspond [1] M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton, to the tuple (2, 1, −1). The tuples corresponding to other and W. Stromquist, On packing densities of permutations, (pairs of) copies of 123 may be verified similarly. Layers Electron. J. Combin. 9 (2002), no. 1, Research Paper 5, 20. MR1887086 퐼ˆ of 7 are highlighted in different colors for easy reference [2] D. Andr´e, Developpments de sec 푥 et de tan 푥, C. R. Acad. throughout Table 2. Sci. Paris 88 (1879), 965–967. [3] D. Andr´e, Mèmoire sur les permutations altern´ees, J. Math. 7 copies tuple copies tuple copies tuple (1881), 167–184. 124,134 (1,0,0) 146,156 (2,1,0) 147,157 (3,1,0) [4] R. W. Barton, Packing densities of patterns, Electron. J. Com- 125,135 (2,0,0) 246,256 (2,1,-1) 247,257 (3,1,-1) 126,136 (3,0,0) 346,356 (2,1,1) 347,357 (3,1,1) bin. 11 (2004), no. 1, Research Paper 80, 16. MR2114184 127,137 (4,0,0) [5] C. B. Presutti and W. Stromquist, Packing rates of measures and a conjecture for the packing density of 2413, Permutation ˆ Table 2. Copies of 123 in 퐼7 = 1325476, which correspond to patterns, London Math. Soc. Lecture Note Ser., vol. 376, tuples (푛, ℓ, 푚) of quantum numbers for the 20 electrons in an Cambridge Univ. Press, Cambridge, 2010, pp. 287–316, atom of calcium. DOI 10.1017/CBO9780511902499.015. MR2732835 [6] M. B´ona, Exact enumeration of 1342-avoiding permutations: We still must show that this mapping of pairs of occur- a close link with labeled trees and planar maps, J. Com- rences of 123 in 퐼ˆ to integer tuples obeys the inequalities bin. Theory Ser. A 80 (1997), no. 2, 257–272, DOI 푘 10.1006/jcta.1997.2800. MR1485138 that determine valid quantum numbers. [7] M. B´ona, Combinatorics of permutations, Discrete Math- ˆ First, by construction, layer 0 of 퐼푘 contains the digit 1, ematics and its Applications (Boca Raton), Chapman & 푘−1 while for 퐿 ≤ , layer 퐿 contains the digits 2퐿 and 2퐿+1. Hall/CRC, Boca Raton, FL, 2004. With a foreword by 2 푘 Richard Stanley. MR2078910 When 푘 is even, layer contains the digit 푘. 2 [8] A. Burstein, P. Hästö, and T. Mansour, Packing patterns Consider the occurrence 휋푎휋푏휋푐 of 123 in 퐼ˆ푘. Since 휋푏 into words, Electron. J. Combin. 9 (2002/03), no. 2, Re- search paper 20, 13. Permutation patterns (Otago, 2003). comes from layer ℓ + 1, we know ℓ ≥ 0 and 휋푐 comes MR2028289 from layer ℓ + 2 or higher. This implies 휋푐 ≥ 2(ℓ + 2). Together with the fact that 푛 = 휋 − ℓ − 3, we have 푛 ≥ [9] P. Erd˝osand G. Szekeres, A combinatorial problem in geom- 푐 etry, Compositio Math. 2 (1935), 463–470. MR1556929 2(ℓ + 2) − ℓ − 3 = ℓ + 1 ≥ 1. 푛 ≥ ℓ + 1 also implies [10] I. M. Gessel, Symmetric functions and P-recursiveness, J. ℓ ≤ 푛 − 1. Finally, since |푚| denotes the layer of 휋푎 and Combin. Theory Ser. A 53 (1990), no. 2, 257–285, DOI ℓ + 1 denotes the layer of 휋푏, we have |푚| + 1 ≤ ℓ + 1, or 10.1016/0097-3165(90)90060-A. MR1041448 |푚| ≤ ℓ, as desired. These are exactly the restrictions on [11] P. A. Hästö, The packing density of other layered per- quantum numbers discussed earlier! Notice that a new ℓ mutations, Electron. J. Combin. 9 (2002/03), no. 2, Re- value is introduced for each even value of 푘 since 퐼ˆ2푖 has search paper 1, 16. Permutation patterns (Otago, 2003). MR2028271 one more layer than 퐼ˆ2푖−1. This corresponds exactly to the frequency with which new subshell shapes are introduced [12] D. Knuth, The Art of Computer Programming: Volume 1, Addison-Wesley, 1968. in the periodic table. [13] G. Miessler and D. Tarr, Inorganic Chemistry, 2nd edition, Conclusion Prentice Hall, 1998. [14] OEIS Foundation Inc., The On-Line Encyclopedia of In- This connection between pattern packing and physical teger Sequences, 2019, oeis.org/A168380. chemistry is striking even to long-time permutation pat- [15] A. L. Price, Packing densities of layered patterns, ProQuest terns researchers. Similarly, the quasi-polynomial se- LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of quence obtained for 휈(123, 퐼ˆ푘) had no previous interpre- Pennsylvania. MR2695616 tation in the literature other than as a sequence of atomic [16] R. P. Stanley, Catalan numbers, Cambridge University numbers. What, if any, chemical interpretation is there Press, New York, 2015. MR3467982

1000 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 FEATURED TITLES FROM THE [17] W. Stromquist, Packing layered posets into posets, preprint, 1993. Available at walterstromquist.com EUROPEAN /publications.html. [18] D. Warren, Optimal packing behavior of some 2-block MATHEMATICAL patterns, Ann. Comb. 8 (2004), no. 3, 355–367, DOI 10.1007/s00026-004-0225-3. MR2161643 SOCIETY [19] D. Warren, Optimizing the packing behavior of layered per- mutation patterns, ProQuest LLC, Ann Arbor, MI, 2005. The- sis (Ph.D.)–University of Florida. MR2707381 K 3 Surfaces [20] D. Warren, Packing densities of more 2-block patterns, Shigeyuki Kondo¯, Nagoya Adv. in Appl. Math. 36 (2006), no. 2, 202–211, DOI University, Japan 10.1016/j.aam.2005.06.004. MR2199989 K 3 surfaces are a key piece in the classification of complex analytic or algebraic surfaces. The term was coined by A. Weil in 1958—a result of the ini- tials Kummer, Kähler, Kodaira, and the mountain K2 found in Karakoram. The most famous example is the Kummer surface discovered in the 19th century. K 3 surfaces can be consid- ered as a 2-dimensional analogue of an elliptic curve, and the theory of periods—called the Torelli-type theorem for K3 surfaces—was established around 1970. Since then, Lara Pudwell several pieces of research on K 3 surfaces have been under- taken and more recently K 3 surfaces have even become of Credits interest in theoretical physics. Opening graphic is courtesy of vchal via Getty. This book is an introduction to the Torelli-type theorem for complex analytic 3 surfaces and its applications. The Figures 1–3 and 5 are courtesy of Lara Pudwell. K theory of lattices and their reflection groups is necessary Figure 4 is courtesy of Lara Pudwell. Derived from work by to study K 3 surfaces, and this book introduces these Ivan Griffin available at www.texample.net/tikz notions. In addition to lattices and reflection groups, /examples/periodic-table-of-chemical the book contains the classification of complex analytic -elements/. surfaces, the Torelli-type theorem, the subjectivity of the Photo of Lara Pudwell is courtesy of Valparaiso University. period map, Enriques surfaces, an application to the moduli space of plane quartics, finite automorphisms of K 3 surfaces, Niemeier lattices and the Mathieu group, the automorphism group of Kummer surfaces and the Leech lattice. This book will prove helpful to researchers in algebraic geometry and related areas and to graduate students with a basic grounding in algebraic geometry. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

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