Natural Sorting over Permutation Spaces

By R. M. Baer and P. Brock

0. Introduction. In this paper we continue the study, begun in [1], of some combinatorial problems related to monotonicities that occur in certain spaces of finite sequences. These spaces are equipped with standard probability measures, so that one may study the distribution of monotonicities in such spaces and, in particular, the expected lengths of maximal monotonie subsequences. These, in turn, are upper bounds on the expected lengths of monotonie subsequences obtained by applying some selection process over the space of sequences. The problem which we have called natural sorting is concerned with the maximization of these expected lengths. The scheme of the paper is as follows. In Section 1 we give definitions and review some background material. In Section 2 we describe the distribution of maximal sequences occurring in the space of permutation sequences.* These distributions have been computed exactly for spaces of permutations of length n = 2(1)36, and have been approximated by Monte Carlo computations for certain values of n ranging up to 10,000. In Section 3 we consider several selection strategies and the corresponding distribution of selected monotonie subsequences. The computations were performed on the IBM 7094 at the Computer Center of the University of California, Berkeley. We are indebted to David M. Matula for some of the calculations in Section 3. We should like to thank Geri Stephen for her assistance in the preparation of the manuscript.

1. Definitions and Conventions. 1.1. Throughout this paper the term sequence is to be understood to mean finite sequence. Standard terms, if not here defined, are used according to the definitions given in [3]. 1.2. Definition. Let (X, <) be a (totally) ordered space and let n he a fixed natural number. A partial ordering is induced in each element of the cartesian product Xn in the following natural way. If (xi, • • •, x») is in Xn let S denote the space consisting of the set {xi, ■■ -, Xn] together with the partial ordering x; ^ x¡ if and only if x¿ < x¡ and ¿ < j. In what follows, the space S varies over the set of permutation sequences obtained from (1, 2, • • -, n). 1.3. Definition. A chain in *Sis a (totally) ordered subset of S. The length of a chain is the number of elements in it. A maximal chain in S is a chain that is not a proper subset of any other chain in S. A maximal chain in S which has length at least as great as that of any other chain in S is called a spine of S. 1.4. Definition. A maximal chain in S which has length at least as small as that of any other chain in S is called an Erdös chain. (For results on Erdös chains, see [1], [8], and [12].)

Received February 6, 1967. * We shall treat cases of other sorting spaces—in particular sorting spaces of binary sequences —separately. 385

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1.5. Definition. By an nth order selection algorithm (applied to sequences of length n) is meant an algorithm A which selects a monotonie subsequence from a sequence (xi, • • -, xn) according to the following scheme. The first entry xi is selected or re- jected on the basis of its size according to a rule of A, so we might write, according to A(xi). The second entry is selected or rejected in a manner determined by the ordered pair (xi, X2) so we might write, according to A(xi, x2); and here we are to understand that A acts upon knowledge of whether Xi was selected or not. Sim- ilarly x3 is selected or rejected by A according to A(xi, x2, x3), and here we under- stand that A has the information as to which (if any) subset of {xi, x2j was se- lected. And so, for each x,-, the selection or rejection of x, is determined by A on the basis of the set \x¡ :j ^ ¿} and upon the selected subset of this set. So an nth order selection algorithm is essentially one in which all information (concerning selections already made and the preceding part of the original sequence) up to the current point of selection is available to the algorithm. The opposite notion is embodied in the definition of Oth order selection algorithm, which, applied to a se- quence (xi, • • -, x„) selects or rejects each x¿ (i ^ n) according to A(x,-, s) where s is the value of the last selection (if there were such) preceding the decision to select Xi) in other words, a Oth order algorithm simply selects or rejects each xt according to a rule which depends only upon this Xi and the last previously selected entry and is oblivious to the past history of the sequence. 1.6. Definition. The distinguished (nondecreasing) subsequence of the sequence (xi, • • -, Xn) is obtained using the Oth order selection algorithm: Select X\. For each ¿ > 1, select x¿ if x, is not less than the element selected last, prior to con- sideration of Xi.

2. The Distribution of Monotonicities in Pn. In [1] the authors considered the question of monotonicities in the space Pn consisting of the space of all permutations of (1, 2, • ■-, n). Our viewpoint was to consider simultaneously the maximal in- creasing and decreasing subsequences of the elements of this space. Some prelim- inary results concerning this distribution were obtained in [1], and an encompassing result was derived by Schensted [12], who showed the relationship between the dis- tribution and representations of the symmetric group. However, the question of the actual distributions was left open both by [1] and [13]. We now proceed to fill in this gap by exhibiting the results of the requisite computations. First, however, we review the basis of the calculations. 2.1. Definition. A Young tableau of order n is an array of the integers 1,2, • • -, n satisfying the following. The array consists of rows and columns. For each row, the entries in that row form an increasing sequence. For each column, the entries (moving down that column) form an increasing sequence. Each row contains at least as many entries as the row beneath it. Each column contains as least as many entries as the column to its right. Each permutation of (1, 2, • • -, n) uniquely determines a Young tableau. The determination proceeds as follows. Let the permutation be (xi, x2, • ■-, x„). For the moment, define the first entry in the first row of the tableau to be x\. Now, if at the ¿th step, the first i entries of the sequence have been used in the developing tableau then at the next step the element x¿+i is inserted into the first row of the tableau by displacing the smallest entry in the first row which is larger than x,+i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NATURAL SORTING OVER PERMUTATION SPACES 387

or by appending xi+i at the end of the first row if it is larger than all entries in the first row. If an entry y is displaced from the first row by xi+i then y is inserted into the second row by letting it displace the smallest entry in the second row which is larger than y or by simply appending y to the second row if there is no such ele- ment. The process is continued from row to row until either the original xi+i or a displaced element is appended to the end of a row. Then the whole process is re- newed for Xi+2, ■• ■ until all of the entries of the original permutation sequence have been entered into the tableau. 2.2. Theorem (Schensted). If T is the Young tableau generated by the permuta- tion sequence (xi, • • •, x») then the greatest length of a maximal monotone increasing subsequence of (xi, ■■ ■, xn) is equal to the number of columns of T, and the greatest length of a maximal monotone decreasing subsequence of (xi, • • -, xn) is equal to the number of rows of T. 2.3. Definition. By a partition of the positive integer n we mean a monotonically nonincreasing sequence of positive integers which sum to n. By a partition tableau corresponding to a partition (mi, • • •, mk) we mean an upper-left rectangular array of cells (for example, the figure below)

in which there are mi cells in the first row, m2 cells in the second row, • • •, and mk cells in the kth row. To each cell of a partition array is assigned a number h called its hook number. If b is the number of cells below a designated cell and if r is the number of cells to the right of this designated cell, then the value of h corresponding to the designated cell isft = &4-r+l. We illustrate these matters in 2.4. 2.4. Example. Consider the four sequences: (i) 2, 1, 5, 4, 3, 6 (ii) 3, 2, 1, 6, 5, 4 (iii) 3, 1, 4, 2, 5, 6 (iv) 6, 5, 4, 3, 2, 1 According to our convention, these sequences correspond to the four partially ordered sets with Hasse diagrams [3] :

(i) (ii) (iii) (iv)

The corresponding Young tableaux are:

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(i) (ii) (iii) (iv)

5 6

The Young tableaux with their associated hook numbers are:

(i) (ii) (iii) (iv)

w = [61/5-32]2 w = [6!/4-32-22]2 w = [61/5-4-22]2 w = 1 = 162 = 52 = 92

If one considers the Young tableaux generated by permutations of (1, 2, ■• -, n) it is evident that different permutations can give rise to the same tableau shape (i.e., partition tableau). The number of tableaux with the same shape arising in this fashion is given by a powerful combinatorial theorem : 2.5. Theorem (Frame-Robinson-Thrall [10]). The number of tableaux with given shape that contain the integers 1,2, ■ ■ ■,n is n\/ \\ ¡hi, where the hi are the hook numbers associated with the cells of the tableau. Finally we need [12, Theorem 3], which is obtained from 2.2 and 2.5. 2.6. Theorem (Schensted). The number of sequences consisting of the distinct numbers xi, ■ ■ ■, xn and having a longest increasing subsequence of length j and a longest decreasing subsequence of length k is the sum of the squares of the numbers of identically shaped partition tableaux with j columns and k rows. Based upon the preceding results, the distribution of monotonicities has been computed exactly for the spaces pn (n ^ 36). The procedure is to generate the dis- tinct partitions of n, to generate the partition tableaux, load the cells with their hook numbers, evaluate the square of the value of the Frame-Robinson-Thrall function. This result is then added to one of several running sums, corresponding to the maximal partition element mi (for the increasing subsequences of greatest length where this length happens to be m/) or to one of several running sums cor- responding to fc (for the case of the decreasing subsequences of greatest length where this length happens to be k, and (mi, • ■ -, mk) is the current partition of n), or to one of several running sums corresponding to j = max [m, k] for the case of

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NATURAL SORTING OVER PERMUTATION SPACES 389 monotone subsequences of greatest length when this length happens to be j. These exact distributions were calculated on a 7094 using the multiple-precision fixed point routines described in [2]. The results are shown in Table 1, giving the distribution of permutations of (1, 2, - • -, n) which contain increasing subsequences of greatest length as well as the distribution according to monotonie subsequences of greatest length. According to the statement of a celebrated theorem of Erdös [8], every sequence of length k2 + 1 contains a maximal monotone subsequence of length at least k + 1. Hence the zeros as the first k entries in Table lb. The exact calculation for the distribution of monotonicities could be carried out on the 7094 only for sequences of length ;£ 36. For sequences of greater length, we examined the expected length of the monotonically increasing subsequences of greatest length by a Monte Carlo procedure. That is, using a pseudo-random number generator we generated sequences of real numbers x (0 < x < 1) with uniform distribution, and verified that the distribution for sequences of length S 36 matched the exact distributions. We then examined the distributions in Monte Carlo fashion for sequences of length up to 10,000. The results are summarized in Fig. 1 (and Table 2). It may be observed that the expected length of the spine, over the range n ^ 10,000, approaches the value Ln = 2 V n. This gives us a standard for estimating efficiencies of sorting algorithms which may be applied over Pn.

IOO

10,000

Figure 1. Mean values of lengths of monotone increasing subsequences of greatest length over the space of permutations of length n, using Monte Carlo trials on sequences of pseudo- random x, 0 < x < 1.

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License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 406 K. M. BABR AND P. BROCK

3. Two Sorting Strategies over Pn. We consider first the 0th order sorting strategy over Pn. This consists of selecting the initial element (in a permutation sequence), and, having selected an element, selecting the first following element in the sequence which is greater. The resulting selected subsequence is called the distinguished subsequence. 3.1. Theorem. In the space Pn the expected length of the distinguished subsequence is

Ln=Y, AT1. k=l Proof. The statement of the theorem is clearly true for n = 1. We proceed by induction. Suppose that the statement of the theorem holds for k á n. Given any particular sequence in Pn, there are n + 1 waysin which sequences of Pn+i may be formed through the adjunction of the number n + 1 to this particular sequence, and with each of these new sequences is associated the probability (n + l)~l. For each new sequence so constructed, the corresponding distinguished subsequence ends with the adjoined number. Hence the expected length of the corresponding distinguished subsequence is greater by unity than the expected length of a dis- tinguished subsequence in that part of the original sequence preceding the adjoined integer, averaged over the positions that the adjoined integer may occupy. This is just n Ln+i = (n+ I)"1 £ Lk + 1 k-0 where L0 is taken to be zero. It follows that Ln+l = Ln+(n+ l)-1. This concludes the proof. We now consider the nth order strategy for natural sorting in Pn. According to the definition, the sorting algorithm of an nth order strategy has available, at the point where it decides whether to select the ith sequence element, the identity of the previously considered sequence elements xk, k < i. The selection algorithm for Pn accordingly has the following form. A choice level Ci (where Ci is a natural number, 0 < Ci ¿ n) determines the selection of the first sequence element xi (i.e. if xi g Ci then xi is selected). A selection level C2 = fi(xî) is defined, depend- ing upon xi. The second sequence element x2 is selected accordingly (if ii á i¡ S C2 in the case that xi was selected, or else simply x2 i= C2 in the alternate case). What is required is the sequence of functions fo(S) = Ci, fi(S) = C2, • ■• (where S is an arbitrary sequence in Pn) which maximizes the expected length of the selected subsequence. To show how the choice levels are made, we will consider the example of Pa.. We suppose that the expected lengths from the optimal choice functions are known for Pi, Pi, and P3. (These expected lengths are: (Li> = 1, (L2) = 3/2, (Li) = 2.) There are four possibilities for the value of Ci (i.e. Ci = 1, 2, 3, or 4). Consider the expected lengths of the selected subsequence which results from each possible value of Ci. Thus, if Ci = 1 the first element of a permutation sequence will be selected if and only if it has the value unity. In this case the remaining sequence consists of a permutation of the sequence (2, 3, 4) and by, assumption, the strategy

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NATURALsorting over permutation spaces 407

and expected length of P3 are known and may be applied to the permutation of (2, 3, 4)-—with certain obvious adjustments. The expected length in this case, which we shall denote ¿4(1), is just the value of L3 + 1 weighted by the probability (which is 1/4) that unity occurs as the initial entry of the sequence from Pi added to the value (L3) which in turn is weighted by the probability that unity does not occur as the first sequence entry. So, L¿1) = 4_1[(1 + (L3)) + 3(L3>] .

Table 2 Expected lengths of (1) monotonically increasing subsequences of greatest length, (2) of monotone subsequences of greatest length, and (3) observed (Monte Carlo) means of monotonically increasing subsequences of greatest length, (4) of mono- tone increasing subsequences selected according to the nth order strategy; (5) is the initial selection level used in (4). JV_il)_(10_(S)_ij)_05) 3 2.000 2.333 4 2.416 2.916 5 2.791 3.300 2.73 3 6 3.140 3.650 3.155 3.04 3 7 3.465 4.021 3.446 3.33 3 8 3.770 4.350 3.760 3.60 4 9 4.059 4.647 4.049 3.86 4 10 4.334 4.938 4.333 4.10 4 11 4.598 5.222 4.577 4.32 4 12 4.852 5.490 4.849 4.54 5 13 5.096 5.745 5.115 4.75 5 14 5.332 5.991 5.323 4.95 5 15 5.561 6.232 5.560 5.15 5 16 5.783 6.465 5.786 5.33 5 17 5.999 6.691 6.013 5.51 5 18 6.209 6.910 6.213 5.69 6 19 6.414 7.123 6.417 5.86 6 20 6.614 7.332 6.618 6.03 6 21 6.810 7.536 6.832 6.19 6 22 7.002 7.736 7.002 6.35 6 23 7.189 7.931 7.165 6.50 6 24 7.373 8.122 7.354 6.65 7 25 7.554 8.309 7.556 6.80 7 26 7.731 8.493 7.741 6.94 7 27 7.905 8.673 7.926 7.09 7 28 8.076 8.851 8.074 7.23 7 29 8.244 9.025 8.258 7.36 7 30 8.410 9.196 8.420 7.50 7 31 8.573 9.365 8.553 7.63 8 32 8.734 9.531 8.717 7.76 8 33 8.892 9.695 8.871 7.87 8 34 9.049 9.857 9.026 8.01 8 35 9.203 10.016 9.206 8.14 S 36 9.355 10.173 9.368 8.26 8 100 16.723 14.05 14 200 24.508 20.02 20 1000 58.154 44.99 44 10000 192.2 142.07 141

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If we take Ci = 2 then the argument proceeds in a similar way, except that now if either 1 or 2 occurs as the initial sequence element it is selected, and if 2 occurs and is selected then since the selection algorithm applies now only to se- quence entries with values greater than 2, the strategy and expected length of the selected subsequence from P2 comes into play, thus L4(2) = 4->[(l + (Ls>) + (1 + (Z.2» + 2]. And similarly L4(3) and L4(4) may be evaluated. Then Ci is defined as that value of A;which maximizes Li(k). In the general case

(Ln+i) = max in + l)"1 k + On + 1 - k)(Ln) + ¿ (L„-,-+1> 0

Conclusions. Over the range which has been examined (n ^ 10,000), let n be the ratio of the expected value of a monotone subsequence, selected according to the 0th order strategy, to the expected length of the monotone subsequence of greatest length. Then Xîfc"1 l°gn n 1 2Vn 2Vn Let r2 be the ratio of the expected length of the monotone subsequence, selected according to the nth order strategy, to the expected length of the monotone sub- sequence of greatest length. Then r2 ~ (2n)l/2/2Vn = 1/V2 .

Table 3 Tableaux with maximal weights for group of order n; 10 g n ^ 36

order 10 order 11 order 12 order 18 order 14 order 15 7531 86421 96421 96431 10 75421 97531 531 531 631 7421 7421 7531 31 31 41 41 41 531 11 2 2 2 3 1 111 1 order 16 order 17 order 18 order 19 order 20 10 86421 1186421 12 975321 12 975421 12 10 75421 7531 8531 8531 96421 9 7421 531 631 631 631 641 3 1 4 1 4 1 4 1 4 2 1 2 2 2 3 1 1111

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NATURAL SORTING OVER PERMUTATION SPACES 409

Table 3—Continued

order 21 order 22 order 23 order 2J+ 13 107 54 2 1 12 10 8 6421 131086421 1411975321 10 7421 97531 10 7531 10 7531 741 7531 8531 8531 52 531 631 631 4 1 3 1 4 1 4 1 2 12 2 1 1 1 order 25 order 26 order 27 order 28 14 11 975421 15 11 975421 15 12 975421 15 12 976421 11 86421 12 86421 12 96421 12 96431 8531 9531 9631 10 7421 631 731 741 741 4 1 5 1 5 2 5 2 2 3 4 1 4 1 12 2 2 1 1 1 order 29 order 30 order 31 14 12 10 8 6421 15121086421 161311975321 11 97531 12 97531 12 97531 97531 10 7531 10 7531 7531 8531 8531 531 631 631 3 1 4 1 4 1 12 2 1 1 order 32 order 33 order 34 13 16 11 9 7 5 4 2 1 17 13 11 9 7 5 4 2 1 17 14 11 9 7 5 4 2 1 13 10 8 6 4 2 1 14 10 8 6 4 2 1 14 11 8 6 4 2 1 10 7531 11 7531 11 8531 8531 9531 9631 6 3 1 7 3 1 7 4 1 4 1 5 1 5 2 2 3 4 1 1 2 2 1 1 order 35 order 36 17 14 11 9 7 6 4 2 1 17 14 12 9 7 6 4 2 1 14 11 8 6 4 3 1 14 11 9 6 4 3 1 12 96421 12 97421 9 6 3 1 9 6 4 1 7 4 1 7 4 2 5 2 6 3 1 4 1 4 1 2 2 1 1

Open questions. This study has raised some questions of which the following, in the area of asymptotic combinatorics, are particularly interesting:

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 410 R. M. BAER AND P. BROCK

(1) What is the asymptotic form of the distribution of spines in the case of permutation spaces? Presumably an analytical derivation of the answer to (1) would require an answer to (2) For given n, which Young tableau maximizes the value of the Frame- Robinson-Thrall function? If the maximizing tableau is denoted Tn then (3) Is there an algorithm which permits the immediate construction of Tn+i from TJ In computing the exact distributions of the spines (for n ^ 36), the correspond- ing Tn were obtained automatically. These are exhibited (for 10 ^ n ^ 36) in Table 3, although it is not believed that they suggest the form of Tn for large n.

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1. R. M. Baer & P. Brock, "Natural sorting," /. Soc. Indusl. Appl. Math., v. 10, 19G2, pp. 284-304. MR 27 #4396. 2. R. M. Baer & M. G. Redlich, "Multiple-precision arithmetic and the exact calculation of the 3-i, &-jand 9-j symbols," Comm. ACM, v. 7, 1964, pp. 657-659. MR 31 #865. 3. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc, Providence, R. I., 1940; rev. ed., 1967. 4. L. Carlitz, "A binomial identity arising from a sorting problem," SI AM Rev., v. 6, 1964, pp. 20-30. MR 29 #4708. 5. L. Carlitz, "Some multiple sums and binomial identities," /. Soc. Indust. Appl. Math., v. 13, 1965, pp. 469-486. MR 31 #2160. 6. P. Erdös, "On some asymptotic formulas in the theory of partitions," Bull. Amer. Math. Soc, v. 52, 1946, pp. 185-188. MR 7, 273. 7. P. Erdös & J. Lehner, "The distribution of the number of summands in the partitions of a positive integer," Duke MathJ., v. 8, 1941, pp. 335-345. MR 3, 69. 8. P. Erdös & G. Szekeres, "A combinatorial problem in geometry," Compositio Math., v. 2, 1935, pp. 363^70. 9. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed., Wiley, New York, 1957. MR 19, 466. 10. J. S. Frame, G. de B. Robinson & R. M. Thrall, "The hook graphs of the symmetric group," Canad. J. Math., v. 6, 1954, pp. 316-324. MR IS, 931. 11. Hansraj Gupta, C. E. Gwyther & J. C. P. Miller, Tables of Partitions, Royal Society Mathematical Tables, Vol. IV, Cambridge Univ. Press, New York, 1962. 12. C. Schensted, "Longest increasing and decreasing subsequences," Canad. J. Math., v. 13, 1961, pp. 179-191. MR 22 #12047. 13. M. P. Schutzenberger, "Quelques remarques sur un construction de Schensted," Math. Scand., v. 12, 1963, pp. 117-128. MR 32 #7433. 14. G. Szekeres, "An asymptotic formula in the theory of partitions," Quart. J. Math. Oxford Ser. (2), v. 2, 1951, pp. 85-108. MR 13, 210.

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