Reconstructing permutations from identification minors

We consider the problem whether a permutation of a finite set is uniquely determined by its identification minors. While there exist non-reconstructible permutations of every set with two, three, or four elements, we show that every permutation of a finite set with at least five elements is reconstructible from its identification minors. Moreover, we provide an algorithm for recovering a permutation from its deck. We also discuss a generalization of this reconstruction problem, as well as the related set-reconstruction problem.


Introduction
Reconstruction problems are a general class of mathematical problems that concern whether a mathematical object is uniquely determined by pieces of partial information about the object.In this paper, we will discuss reconstruction problems that fall into a general framework that was elegantly formalized by Couceiro, Schölzel, and the current author in [4,Section 2.2].Roughly speaking, a reconstruction problem comprises a set O of objects, a way of forming from each object O ∈ O certain derived objects called the cards of O, and an equivalence relation on O. Once this data is specified, we may ask whether, or to what extent, an object is uniquely determined, up to equivalence, by its deck, i.e., the collection of (the equivalence classes of) its cards.
Note that a given object may give rise to the same derived object, up to equivalence, in many different ways, and it is important to keep track of the number of times each card appears in the deck.In other words, by a "collection of cards" we mean a multiset the electronic journal of combinatorics 22(4) (2015), #P4.20 of cards.Ignoring the multiplicities, taking "collection" to mean a set, we are dealing with a somewhat different yet related problem, often referred to as a set-reconstruction problem.
Perhaps one of the most renowned reconstruction problem comes from graph theory.Is every graph with at least three vertices uniquely determined, up to isomorphism, by the collection of its one-vertex-deleted subgraphs?It was conjectured by Kelly [7] and Ulam [15] that the answer to this question is positive, but this still remains as an open problem that has received considerable attention over the years.
Analogous reconstruction problems have been formulated for many other types of mathematical objects, such as directed graphs, hypergraphs, relations, posets, functions, matrices, matroids, integer partitions, and so on.The main topic of this paper is reconstruction problems the objects of which are permutations.
Evidently, there are many possibilities for defining how the cards of a permutation are formed, and various reconstruction problems involving permutations have accordingly been formalized in the literature.For instance, Ginsburg [6], Raykova [13], and Smith [14] have studied the reconstruction problem where, for a fixed parameter k ∈ N + , the cards of a permutation π = π 1 π 2 . . .π n are the so-called (n − k)-minors of π, i.e., the patterns of the subsequences of π of length n − k.For example, the patterns of the subsequences of length 3 of the permutation 31524 are 213,312,213,231,132,213,132,132,123,312.
In this paper, we study yet another reconstruction problem of permutations.In this variant, the cards of a permutation are its identification minors.The identification minors of π = π 1 π 2 . . .π n are formed, for each two-element subset {i, j} (i < j) of {1, . . ., n}, by replacing the entry j by i, deleting the second occurrence of i, and taking the pattern of the resulting sequence.For example, the identification minors of 31524 are 1423, 3124, 2143, 3142, 3124, 2143, 3142, 3124, 3142, 3142.The identification minors of permutations are in a complete analogy to the identification minors of functions of several arguments.The formation of minors is a way of deriving new functions from a given one that has great significance in universal algebra.A function f : A n → B is said to be a minor of another function g : A m → B, if f can be obtained from g through a combination of the following operations: introduction of inessential arguments, deletion of inessential arguments, identification of arguments, and permutation of arguments.In the special case when f is obtained from g through the identification of a single pair of arguments, f is called an identification minor of g.Examples of recent work on minors of functions include the papers by Couceiro and Foldes [1], Couceiro, Schölzel, and the current author [2], Ekin, Foldes, Hammer, and Hellerstein [5], Pippenger [12], Wang [16], Willard [17], and Zverovich [18].
A reconstruction problem for functions and identification minors was formulated in [8]: the objects are functions A n → B, the cards of a function f : A n → B are its identification minors, and two n-ary functions are considered equivalent if each one can be obtained from the other through permutation of arguments.Several results, both positive and negative, concerning this reconstruction problem were obtained by the current author, partly in collaboration with Couceiro and Schölzel in [3,4,8,9,10].
Certain quotient-like structures of permutations, called "permutations arising from a permutation via partitions", arose as a crucial tool in [10], where the current author studied the reconstructibility of functions determined by the order of first occurrence.(We will discuss this in a bit more detail in Section 6.)The identification minors of permutations, as defined above, constitute an important special case of permutations arising from a permutation via partitions.Against this background, it looks very natural to investigate the reconstruction problem of permutations and identification minors that is the topic of this paper.
This paper is organized as follows.In Section 2, we provide the necessary definitions and we make the immediate observation that if 2 n 4, then there are permutations of {1, . . ., n} that are not reconstructible from identification minor.In fact, if n = 2 or n = 3, then every permutation of {1, . . ., n} is non-reconstructible.In Section 3, we then prove the main result of this paper: for n 5, every permutation of {1, . . ., n} is reconstructible from its identification minors (Theorem 3.7).In fact, the proof provides an effective algorithm for actually recovering a permutation from its deck; this is explained in Section 4. In Section 5, we discuss the related set-reconstruction problem and observe that not every permutation of {1, . . ., n} is set-reconstructible, for all n 2, i.e., if the multiplicities of cards are ignored, then reconstruction is no longer possible in general, no matter how large n is.Finally, in Section 6, we make some conclusive remarks and suggest a natural generalization of the reconstruction problem discussed in this paper.Analysis of this generalization remains a topic of further research.

Identification minors of permutations 2.1 Notation
We will first introduce some general notation which we will use throughout the paper.The set of positive integers is denoted by N + .For n ∈ N + , the set {1, . . ., n} is denoted by [n].The symmetric group of [n] is denoted by Σ n .Any permutation π ∈ Σ n corresponds to the sequence (or word) π 1 π 2 . . .π n , where π i = π(i) for every i ∈ [n].If π(i) = j, then we say that j is the location of i in π, or that i is at the j-th position in π.We write i < π j to denote the fact that i is located to the left of j in the sequence π 1 π 2 . . .π n , i.e., π −1 (i) < π −1 (j).The pattern of a sequence a 1 a 2 . . .a of integers with no repeated entries is the unique permutation π = π 1 π 2 . . .π ∈ Σ such that a 1 a 2 . . .a is order-isomorphic to π 1 π 2 . . .π , i.e., the relative order of elements is the same in both sequences.
The set of all 2-element subsets of [n] is denoted by [n]  2 .For a permutation π ∈ Σ n and a 2-element set I = {i, j} ∈ [n]  2 with i < j, let π I be the permutation of [n − 1] that is obtained from the sequence π 1 π 2 . . .π n by performing the following steps: 1. Replace j by i and delete the rightmost occurrence of i in the resulting sequence.
the electronic journal of combinatorics 22(4) (2015), #P4.20 (Equivalently, if j < π i then swap i and j.Delete j from the resulting sequence.)2. Decrease any number greater than j in the resulting sequence by 1. (Equivalently, take the pattern of the resulting sequence.) The permutations of the form π I for some I ∈ [n]   2 are called identification minors of π.

Reconstruction problem
Now we can formulate a reconstruction problem for permutations and identification minors.The objects are permutations π ∈ Σ n for some n ∈ N + .The cards of π ∈ Σ n are the 2 ).The equivalence relation on Σ n is the equality relation.The deck of π is the multiset {π I : I ∈ [n]  2 } of the identification minors of π, and it is denoted by deck π.If τ ∈ Σ n and deck π = deck τ , then τ is a reconstruction of π.If every reconstruction of π is equal to π, then π is reconstructible.
If 2 n 4, then not all permutations of [n] are reconstructible.For n = 2, this is obvious, because there are two permutations of {1, 2} but only one permutation of {1}; hence both permutations of {1, 2} necessarily have the same deck.For n = 3, a simple counting argument shows that there must exist permutations with the same deck: there are 3! = 6 permutations of {1, 2, 3}, each has 3 2 = 3 cards that are permutations of {1, 2}.The number of permutations of {1, 2} is 2! = 2.The number of 3-element multisets over a 2-element set is 4. Therefore, the number of possible decks is less than the number of permutations of {1, 2, 3}, so the non-reconstructibility of some permutations is unavoidable.Unfortunately, the same counting argument will not work when n 4, so we need to take a different approach.The case n = 4 is still quite easy to work out by hand.Indeed, in Table 1, we have enumerated all permutations of [n] and their identification minors, for 2 n 4. Permutations with identical decks are marked with * and †.It can be read off from the table that the permutations 1342 and 1423 have the same deck.The table also reveals the fact that, in fact, no permutation of {1, 2, 3} is reconstructible.
These observations raise the question whether permutations of sets with at least five elements are reconstructible or not.We will address this question in the remainder of this paper.

Reconstructibility of permutations
In this section, we are going to establish the main result of this paper: every permutation of a finite set with at least five elements is reconstructible from its identification minors (Theorem 3.7).First, we need to introduce some notation and establish some auxiliary results.It is an easy exercise to verify the following identities involving binomial coefficients, which will be used frequently in the sequel: . Denote by N (π, , i) the total number of times the element occurs at the i-th position in the cards of π, i.e., the electronic journal of combinatorics 22(4) (2015), #P4.20 It is clear from the definition that the nonempty sets among S π,k 1 , . . ., S π,k 6 constitute a partition of [n]  2 .The motivation for defining this partition is that the location of k in π I depends on the block S π,k i to which I belongs, as described by the following lemma.
, so the element k + 1 at position β is decreased by 1 (so that it becomes k) upon formation of the minor π I .If I ∈ S π,k 1 , then I ⊆ L π (k + 1), so the element that gets deleted upon formation of the minor π I is located to the left from position β, and we conclude that (π 2 ) the element that gets deleted is to the right from position β, and we conclude that (π Consequently, the element k at position α is not decreased nor does it get deleted upon formation of the minor π I , but it may be swapped with before gets deleted.If < π k (i.e., if I ∈ S π,k 3 ), then k and are first swapped before deleting , and we conclude that (π 4 ), then k remains at its place at position α and gets deleted, and we conclude that (π I ) −1 (k) = α whenever I ∈ S π,k 4 .Assume then that I ∈ S π,k 5 ∪ S π,k 6 .Then I [k] and k / ∈ I.This implies that I contains an element strictly greater than k.Therefore the element k at position α is not decreased by 1 upon formation of the minor π I .If I ∈ S π,k 5 , then the element that gets deleted upon formation of the minor π I is located to the left from position α, and we conclude that (π I ) −1 (k) = α − 1. Otherwise (i.e., if I ∈ S π,k 6 ) the element that gets deleted is to the right from position α, and we conclude that (π , and write α := π −1 (k) and β := π −1 (k + 1).
Then the cardinalities of the sets S π,k i (1 i 6) are the following: and the claim about |S π,k 2 | follows.The condition I [k] and k ∈ I is equivalent to the condition that the electronic journal of combinatorics 22(4) (2015), #P4.20 Finally, observe that S π,k 5 ∪ S π,k

2
. Substituting n − k for h π (k) in the latter identity and then applying identity (2), we obtain The latter is not possible, because we are assuming that k ∈ [n − 1], so we conclude that n = α.
Let us write down the minors of σ and τ , providing as much details as we can deduce from the known facts: σ(4) = 6, σ(6) = 7, σ(7) = 5, τ (5) = 7, τ (7) = 6.See the first two columns of Table 4.The rows are labeled as ij, where i, j ∈ [7] with i < j.The row then represents the minor σ I with I = {σ(i), σ(j)} or τ I with I = {τ (i), τ (j)}.In the bottom part of the tables we have counted the number of times each element of [6] occurs at each position in the table.The blanks in the table represent unknown values.Note that N (σ, c, 6) = 0 for every c ∈ {1, 2, 3}.
Therefore, it only remains to consider the case τ (6) = 4. Having fixed this value, we can complement the information about the cards of τ ; see the rightmost column of Table 4.Each one of the entries marked with * is either 3 or 4; three of them are 3's and three of them are 4's.Exactly which ones are 3 and which ones are 4 depends on which one of the three numbers 1, 2, 3 is mapped to 5 by τ .
We have now established that the conditions of statements (i), (ii), and (iii) are necessary for π −1 (n − 1) = α and π −1 (n) = β.Since all possible values of α, β, and n are covered by statements (i), (ii), and (iii), sufficiency of these conditions will follow if we show that the conditions are mutually exclusive.Observe first that the conditions of statements (ii) and (iii) clearly cannot both hold simultaneously.= 0, i.e., n 2, which contradicts the assumption that n 5. Suppose that (α, β) and (α , β ) are of types (B) and (F), respectively.Then we have n As we have seen above, this yields a contradiction to n 5.
Suppose then that (α, β) and (α , β ) are of types (B) and (H), respectively.Condition C(α , β ) implies that p, q, r are three consecutive integers, i.e., q = p + 1 and r = p + 2. From the conditions on N (π, n − 1, i) for i ∈ {p, q, r} we obtain the following system of simultaneous equations: Rewriting the second equation, using the identity p 2 + p = p+1 2 (see identity (1)), and then substituting n − 1 for p 2 as per the first equation, we obtain n−1 2 + 1 = 2(n − 1), which yields the quadratic equation n 2 − 7n + 8 = 0, which, as we have seen above, has no integer solution, and we have arrived in a contradiction again.
the electronic journal of combinatorics 22(4) (2015), #P4.20  6, filling in the cells in the usual reading order: from left to right and from top to bottom.We conclude that π = 7362514.

Set-reconstructibility of permutations
Having discovered that all permutations of [n] are reconstructible from identification minors whenever n 5, we may ask whether reconstruction is possible if we ignore the multiplicities of cards.As we will see shortly, this is not in general possible, even for large values of n.
The set-deck of a permutation π ∈ Σ n is defined as the set {π I : I ∈ [n] 2 } of its identification minors, and it is denoted by set-deck π.If τ ∈ Σ n and set-deck π = set-deck τ , then τ is a set-reconstruction of π.If every set-reconstruction of π is equal to π, then π is set-reconstructible.Set-reconstructibility clearly implies reconstructibility, but the converse is not necessarily true.
If 2 n 4, then there obviously exist permutations of [n] that are not set-reconstructible, for the simple reason that there are permutations that are not even reconstructible.Further examples of permutations that are not set-reconstructible, despite being reconstructible, can be found by examining Table 1.
By Theorem 3.7, every permutation of a set with at least five elements is reconstructible, but, as the following example illustrates, for every n 3, there exist non-setreconstructible permutations of [n].Investigation of this reconstruction problem for k > 1 remains a topic of future research.For example, the following problem seems natural.For k ∈ N + , let us denote by N k the smallest number N , if one exists, such that for every n N , every permutation of [n] is reconstructible from its (n − k)-identification minors.Theorem 3.7 and the discussion in Section 2 show that N 1 = 5.Does the number N k exist for every positive integer k? Determine N k for k > 1.