On plane and colored partitions of integers

Transcription

1 On plane and colored partitions of integers Fabrice Philippe and Jean-Marie Boé dpt MIAp, Université Paul Valéry, 3499 Montpellier, France and LIRMM, UMR 5506, 6 rue Ada, Montpellier, France (Fabrice.Philippe,Jean-Marie.Boe)@univ-montp3.fr Submitted: Apr 2, 2004 Abstract A new bijective proof of MacMahon s theorem on plane partitions is presented. The correspondence is constructed from the shapes of colored partitions of integers, viewed as preorders in the two-dimensional lattice of positive integers. This correspondence presents several nice features that are lacking in the customary Bender- Knuth correspondence. Similarities between the two correspondences are exhibited, providing geometrical insight into the Robinson-Schensted-Knuth algorithm. Keywords: Plane partitions; Colored partitions; Growth diagram; RSK algorithm; Bender- Knuth correspondence; Sliced permutations Introduction In his famous textbook on combinatorial analysis [6], MacMahon expresses generating series for several classes of plane partitions by means of products. In particular, the generating series for the whole class, depending on the partitioned number, reads (ibid., 495) ( x n ). () n n MacMahon s proof of this fundamental result relies on sophisticated arguments that involve the generating series of several combinatorial objects. Readers familiar with the theory of partitions recognize in () the generating series for certain colored partitions of integers, namely, those where the part n has n available colors. Although MacMahon does not make use of the term, these colored partitions appear in graphical form from the very beginning of his discussion (ibid., 423). These colored integers may indeed be represented by points with two positive integer coordinates (i, j), the n colors available for the integer n being represented by the n points on the line i + j = n. In order to prove the MacMahon s result in a different manner, one

2 shall therefore seek a bijective correspondence between plane partitions of a given integer U and such colored partitions of U. Sixty years passed before Bender and Knuth discovered a bijective proof based on an ingenious adaptation of the Robinson-Schensted-Knuth (RSK) algorithm. In short, colored partitions are viewed as multisets of points. Ordinates are conveniently inserted in a plane partition, while abscissas are recorded into another plane partition having the same Ferrers shape. The two plane partitions are then glued together into a third one. Detailed description and historical references can be found in [, 8]. Turning next to the product form of the generating series for plane partitions with at most r rows and c columns, also presented in MacMahon s book, one may identify the generating series for colored partitions whose geometrical representation fits into a c r rectangle. Besides, the bivariate generating series for plane partitions of U with a trace equal to N, obtained by Stanley [7], is precisely the one for colored partitions of U with N parts. Another point of view deserves to be presented: According to quantum statistics, the product form of the latter generating series is recognized to be the grand-partition function of two-dimensional harmonic oscillators, with U representing the total energy and N the number of particles. Unfortunately, in neither one of the two above examples does the Bender-Knuth (BK) correspondence map to one another the two classes of partitions. In this paper, a bijective correspondence ϕ between colored and plane partitions is constructed that exhibits the nice properties that we are looking for. It explains all the generating series in one stroke. This correspondence has in fact stronger features, one of them being illustrated in Fig.. It also perfectly behaves with respect to transposition. The leftmost plane partition in Fig. is associated by the BK correspondence with the colored partition to its right, which, in turn, is associated by ϕ with the rightmost plane partition. Abscissas increase from left to right, and ordinates increase downwards. Scriptsized numbers are either cumulate margins or diagonal sums Figure : Plane partitions of 73 associated with a colored partition of 73 with 7 parts. Another feature deserves attention: Computing the plane partition shown on the left by using the RSK algorithm is not an easy mental exercise. To the contrary the plane partition shown on the right may be written down by inspection. Indeed, its diagonals are simply the shapes (for the product order) of certain multisets of points extracted from the colored partition. For example, the longest chain in the whole colored partition covers 9 points, 4 more points may be covered with 2 chains, 3 more points with 3 chains, and 4 2

3 chains cover all points, whence the main diagonal (9,4,3,). Likewise, the diagonal (5,3) is the shape of the multiset of 8 points obtained after deleting the two first columns in the colored partition. More generally, these shapes may be computed recursively by adapting local rules due to Fomin [2]. Nevertheless, in spite of dissimilarities between the two correspondences, both in construction and characteristics, they turn out to be very close to one another. Indeed, they follow from two different readings of the same object : The two plane partitions associated with a given colored partition have identical solid graphs (up to an isometry). From this point of view, the correspondence ϕ appears as halfway between the BK correspondence and its composition with the transposition operator. The paper is organized as follows. Useful definitions and notations are given in 2, ending with the main statement about ϕ. The proof is established in 4, once border chains and local growth diagrams have been introduced and studied in 3. Further links with permutations are developed in 5, and relations with the BK correspondence are brought to the fore in 6. 2 Definitions and main statement Let M be the set of all (infinite) matrices with entries in the set N of all nonnegative integers, finitely many of them being positive. The entries of a matrix A are denoted by A i,j, where indices run through N, and the sum of all positive entries of A is denoted by A. A plane partition P is a matrix in M with non-increasing entries in each row and in each column, that is, P i,j max(p i+,j, P i,j+ ). The set of all plane partitions is denoted by P. The support of a matrix A M is the ordered pair (r, c) of integers such that r (c) is the largest of the indices of the rows (columns) containing at least one positive entry. The set of all matrices in M with support (r, c) is denoted by M r,c, and the set P r,c is defined similarly. A colored integer is an ordered pair (n, t) of positive integers, n being called the value and t the color. A colored partition of an integer U is a multiset of colored integers whose values sum up to U. The elements of a colored partition are called its parts. Let the number of available colors for an integer n be denoted by g n. In this paper, we always consider the case where either g n =, the usual (or linear) partitions of integers, or g n = n, which we (somewhat abusively) call colored partitions of integers. A simple geometrical representation of a colored partition consists in associating to each part the point (i, j) of the set L N N such that i + j is the value of the part and j its color. A representing matrix C is obviously obtained by letting C i,j be the number of associated points whose coordinates are (i, j). Conversely, it is clear that any matrix in M represents a unique colored partition. The (linear) partitions of U are commonly represented as non-increasing sequences of non-negative integers summing up to U. In order to lighten notations, a few conventions about sequences are useful. Firstly all the sequences that we consider are finite, but integer sequences are agreed to end up with infinitely many zeros, which are omitted. For instance, the null sequence is denoted by (). Likewise, sequences of sequences are 3

4 extended with null sequences. Secondly the sequence with as k th term and 0 elsewhere is denoted by [k]. Thirdly the shift Sς of a sequence ς = (ς k ) k is the sequence obtained after deleting its first part. Accordingly, ς = k ς k [k], Sς = k ς k+ [k]. The product order on integer sequences is defined component-wise, ς ς k, ς k ς k. Therefore, partitions are those integer sequences λ that satisfy λ Sλ. Recall that the Young lattice Y is the set of all partitions of integers, endowed with the product order. If λ is a partition of U, we write λ = U. A plane partition of U is a plane partition P such that P = U. Plane partitions have been characterized by the property that their rows and columns are partitions. But this may also be done by means of their diagonals, a key point for our purpose. Lemma Let (ς (i) ) i and (τ (j) ) j be two sequences of integer sequences such that ς () = τ (), which diagonally define a matrix P M by { ς (i j+) j if i j P i,j = τ (j i+) i if i j. (2) Then P is a plane partition if, and only if, for all i, j, Proof Sς (i) ς (i+) ς (i) and Sτ (j) τ (j+) τ (j). (3) Immediate from definitions. In the sequel, the set L is regarded as a lattice for the product order. A colored partition is thus also viewed as a preordered multiset of points in L, with p p if both points have the same coordinates (we shall freely switch between the different representations of colored partitions). Let C be a colored partition, and let c k be the maximal cardinality of the union of k chains of C. According with a theorem of Greene [5] the sequence (c k c k ) k is non-increasing. It is thus a partition of the cardinality of C, called the shape of C and denoted by λ(c). Notice that C and its transpose t C are isomorphic preorders, thus have the same shape. The new correspondence ϕ between colored and plane partitions is constructed as follows. First define C (i,j) as the colored partition obtained by deleting in C all the points with abscissa less than i or ordinate less than j. Let λ (i,j) be the shape of this colored partition. For example, λ (,) = λ(c). The matrix with entries λ (i,j) is called the growth diagram of the colored partition C, see Fig. 2. It is worthwhile to point out that, by definition, transposing the growth diagram of C provides with the growth diagram of t C. 4

5 (9, 4, 3, ) (7, 3, 3) (5, 3) (4) (7, 4, 3) (7, 3, 2) (5, 2) (3) (7, 3) (6, 2) (4) () (3, ) (2, ) () () (3) (2) () () Figure 2: Growth diagram of a colored partition We next associate with the colored partition C the matrix ϕ(c) constructed diagonally from the boundary of the growth diagram of C as in (2), that is, { λ (i j+,) j if i j ϕ(c) i,j = λ (,j i+) i if i j. (4) For an example, compare Fig. and Fig. 2. We shall prove that ϕ is a bijective correspondence between colored and plane partitions. It has a much stronger property whose presentation requires us to introduce further definitions. A hook H is a matrix in M where all entries are 0 excepted in its first column and its first row, that is, H i,j = 0 if both i 2 and j 2. The diagonal hook δ(a) of a matrix A is the hook obtained by summing up its entries diagonally, and its cumulative marginal hook γ(a) by cumulating its margins. Formally, for i = or j =, δ(a) i,j = A i+k,j+k, γ(a) i,j = A k,l, (5) k 0 k i l j and the other entries are 0. In Fig., for instance, the diagonal hook of the rightmost plane partition and the cumulative marginal hook of the colored partition are identical. Our main statement follows. Theorem Let H be a hook. The mapping ϕ maps the colored partitions with cumulative marginal hook H to the plane partitions with diagonal hook H bijectively. In particular, ϕ maps a colored partition to a plane partition of identical support, and the number of parts in the former is equal to the trace of the latter. Moreover, ϕ commutes with the transposition operator, as pointed out above. 3 Border chains and local growth diagrams In order to justify our main statement about ϕ, we should primarily know why ϕ(c) is a plane partition and how C is reconstructed from ϕ(c). This can be explained by attentively considering the growth diagram of a colored partition, where particular sequences of partitions play a central role : border chains, defined below. This section is devoted to their study. They will be fully employed in the next section. 5

6 A border chain in Young s lattice is a decreasing sequence (λ (i) ) 0 i n of partitions that satisfies the following conditions (i) For each i 0, λ (i) λ (i+) =. (ii) Let λ (i) λ (i+) = [k i ], the sequence (k i ) 0 i n is non-decreasing. Existence and uniqueness of a border chain with given extremities are studied in the next Lemma. The essential relation with plane partitions and their diagonal characterization (3) is made apparent. Lemma 2 There exists a border chain from λ to λ if, and only if, Sλ λ λ. Such a chain is unique whenever it exists. Proof Assume λ λ, a necessary condition. There is only one sequence of finite sequences λ (i) that satisfies conditions (i, ii) stated above. Roughly speaking indeed, the first part in λ must be decreased one by one down to the first part in λ, and the process must be iterated with the next part. Formally, write λ λ = k (λ k λ k )[k], let s p = p k= (λ k λ k ), and choose i between and λ λ. Since (s p ) in non-decreasing, there is exactly one p i 0 such that s pi < i s pi +. Then the sequence (λ (i) ) 0 i λ λ is defined by λ (0) = λ, λ (i) = λ k p i (λ k λ k)[k] (i s pi )[p i + ]. (6) It is a border chain if, and only if, each λ (i) is a partition. But (6) also reads λ (i) = λ k [k] + λ k[k] + (s pi + i)[p i + ]. (7) For k p i +, λ (i) k only if, k>p i + k p i + λ(i) k+ is either λ k λ k+ or λ k λ k+, thus λ(i) is a partition if, and λ (i) p i + λ (i) p i +2 = λ p i + λ pi +2 + s pi + i 0. Restricting attention to the values i such that i = s pi +, the border chain condition reads λ p i + λ pi +2 0 for all i. To complete the proof, it suffices to notice that for any p out of the set {p i } we have s p = s p+, in which case λ p+ = λ p+ λ p+2 holds trivially. Now the link between border chains and growth diagram of a colored partition is partially disclosed. A local growth diagram is a finite matrix whose entries λ (i,j) are partitions that satisfy the two following conditions: (i) Last row and column are border chains with a common minimum. (ii) For all suitable i, j, entries satisfy the local (Fomin s) rules λ (i+,j) λ (i,j+) λ (i,j) = max(λ (i+,j), λ (i,j+) ), (8) } λ (i+,j) = λ (i,j+) λ (i,j+) λ (i+,j+) λ (i,j) = λ (i,j+) + [k + ]. (9) = [k] 6

7 Here is an example, and the fundamental property of local growth diagrams follows. (6, 3, 2) (5, 3, 2) (5, 2, 2) (5, 2, ) (5, 2) (6, 2, 2) (5, 2, 2) (4, 2, 2) (4, 2, ) (4, 2) (6, 2, ) (5, 2, ) (4, 2, ) (4,, ) (4, ) (6, 2) (5, 2) (4, 2) (4, ) (4) Lemma 3 In a local growth diagram, each row and column is a border chain. Proof Since a transposed local growth diagram is a local growth diagram, it suffices to consider rows. If the local growth diagram has only row or less than 3 columns the result is tautological. In the other cases, we show that the i th row is a border chain only if the (i ) th is. We get notations from the following partial diagram. λ + [k] + [k ] + [k 2] λ + [k] + [k ] λ + [k] λ + [k ] + [k 2 ] λ + [k ] λ Four cases have to be considered, denoted by (x,y) if rule x is first employed then rule y. (8,8) We have k k and k = k first, then k k 2 and k 2 = k 2. (8,9) First k k and k = k, then k = k 2 and k 2 = k 2 +. (9,8) First k = k and k = k +, then k + k 2 and k 2 = k 2. (9,9) First k = k and k = k +, then k + = k 2 and k 2 = k 2 +. In every case, assuming k k 2 0 forces k k 2 0 (the case (9,9) is not consistent with this assumption). Given partitions λ, λ, λ, Lemma 2 implies that there is at most one growth diagram (λ (i,j) ) with support (r, c) such that λ (r,c) = λ, λ (,c) = λ, λ (r,) = λ. If it exists, we are specially interested in its first entry λ (,), denoted by lgd(λ; λ, λ ). In fact, we want to know under what conditions a local growth diagram may be reconstructed from a given triple λ (,), λ (,c), λ (r,). The answer follows. Lemma 4 Given partitions λ and λ, let Λ = {λ Y : max(sλ, Sλ ) λ min(λ, λ )}, M = {µ Y : µ max(λ, λ ), µ = max(λ, λ ), and Sµ min(λ, λ )}. The mapping defined on the set Λ by λ lgd(λ; λ, λ ) is one-to-one and onto the set M. Proof Denoting this mapping by f, it is well defined on Λ by Lemma 2. It is one-to-one because rules (8,9) may be reversed easily: λ (i+,j) λ (i,j+) λ (i+,j+) = min(λ (i+,j), λ (i,j+) ), (0) } λ (i+,j) = λ (i,j+) λ (i,j) λ (i,j+) λ (i+,j+) = λ (i+,j) [k ]. () = [k] 7

8 By Lemma 3 and Lemma 2, if λ Λ and µ = f(λ) then µ max(λ, λ ) and Sµ min(λ, λ ). We next prove, by induction on its support (r, c), that every local growth diagram satisfies λ (,) = max(λ (,c), λ (r,) ). This holds for (r, c) = (, ) trivially. Assume it holds for all (i, j) in L with (, ) (i, j) < (r, c), so that Rules (8,9) force λ (,2) = max(λ (,c), λ (r,2) ), λ (2,) = max(λ (2,c), λ (r,) ). max(λ (,c), λ (r,) ) λ (,) = max(λ (,2), λ (2,) ). Assume λ (,) = λ (,2), the other case is similar. Since λ (,2) is either λ (,c) or λ (r,2) λ (r,), the proof is complete. Finally, consider µ M. By Lemma 2, there is one border chain from µ to λ, and one from µ to λ. Denote them by (λ (,j) ) and (λ (i,) ) respectively. A local growth diagram (λ (i,j) ) may be constructed if rules (0,) operate, which is the case if rule () has not to be employed with k =. Let λ λ (,) in the partial diagram below. λ λ [k ] λ [k ] [k 2 ] λ [k] λ [k] [k ] λ [k] [k ] [k 2] A case study shows that rules (0,) together with our hypotheses k k 2 and kk imply λ (,2) = max(λ (,3), λ (2,2) ) and k k 2 successively. Accordingly, the second row may be computed by iteration, and it is a border chain. Let us iterate the process for successive rows, and suppose we eventually meet a first entry λ (i,j) (i 3, j 2) that cannot be computed because λ (i,j) = λ (i,j ) = λ (i,j ) []. If j = 2, we have λ (i 2,) = λ (i,) + [] since the first column is a border chain, thus λ (i 2,2) = λ (i,) + [] [k] with k (otherwise λ (i,2) was incomputable), whence, by (), λ (i,2) = λ (i,) [k] λ (i,) [], a contradiction. If j 3, a similar contradiction occurs because the (i ) th row is a border chain. Therefore, all entries may be computed. 4 Proof of the main Theorem In the particular case of a permutation matrix, Fomin has given local rules for computing the growth diagram recursively. The partial use of this rules in the previous section is now justified. In the next Theorem, Fomin s construction is generalized to arbitrary matrices. Moreover, when compared to the previous Lemmas the result gives some basic properties of the shapes in the growth diagram of a colored partition. Given a colored partition C, recall that C (i,j) is the colored partition obtained by deleting in C all the points with abscissa less than i or ordinate less than j. Next define Ĉ(i,j) from C (i,j) by deleting the points of coordinates (i, j), if any. The corresponding shape is denoted by ˆλ (i,j). Since every chain in C (i,j) of maximal cardinality contains every point (i, j) if any, the shapes λ (i,j) and ˆλ (i,j) may only differ by their first part. More precisely, ˆλ (i,j) = λ (i,j) C i,j []. (2) 8

9 Theorem 2 Let C M r,c be a colored partition. Its growth diagram (λ (i,j) ) may be computed recursively from formula (2) together with the following ones. (i) For (i, j) (r, c), λ (i,c) = k i C k,c [] and λ (r,j) = l j C r,l []; (ii) For i < r, j < c, ˆλ(i,j) = lgd(λ (i+,j+) ; λ (i+,j), λ (i,j+) ). Proof Since C (i,c) and C (r,j) are chains for all (i, j) (r, c), the last row and column of the growth diagram of C are given by (i). Accordingly, for i < r and j < c, λ (i,c) λ (i+,c) Sλ (i,c) and λ (r,j) λ (r,j+) Sλ (r,j). Therefore, assuming that λ (i+,j+), λ (i+,j), and λ (i,j+) ) are known and satisfy min(sλ (i+,j), Sλ (i,j+) ) λ (i+,j+) max(λ (i+,j), λ (i,j+) ), it suffices to show that ˆλ (i,j) = lgd(λ (i+,j+) ; λ (i+,j), λ (i,j+) ). If this holds, λ (i,j) indeed follows from relation (2); Then by Lemma 4, and remarking that Sλ (i,j) = Sˆλ (i,j), the process may be iterated until λ (,) is computed. To this end, locally expand Ĉ(i,j) in the following way. Let n = Ĉ(i,j) C (i,j+), m = Ĉ(i,j) C (i+,j), c k = k t= C i+t,j, and r l = l t= C i,j+t. Then define a new colored partition E as following. For (k, l) (, ), E n+k,m+l = C i+k,j+l ; For k, c k < l c k, E n+k,l = ; For l, r l < k r l, E k,m+l = ; Elsewhere, E k,l = 0. Fig. 3 shows the local expansion at (2,2) of the colored partition in Fig Figure 3: Locally expanding a colored partition Denote by (µ (i,j) ) the growth diagram of E. By construction, Ĉ (i,j) and E have the same shape, as well as C (i+,j) and E (n+,), C (i,j+) and E (,m+), and C (i+,j+) and E (n+,m+). For < k n +, E (k,m+) obtains from E (,m+) after deleting successive extremal elements that formed a chain. According to a Theorem of Gansner [3], quoted in [2], (µ (k,m+) ) k n+ is thus the border chain from λ (i,j+) to λ (i+,j+). Similarly, 9

10 (µ (n+,l) ) l m+ is the border chain from λ (i+,j) to λ (i+,j+). Next notice that each line to n and column to m in E contain only 0 s but exactly one, and that all entries are 0 in the sub-matrix (E k,l ) (k,l) (n,m). Therefore, according to [2], Fomin s rules (8,9) allow computing (µ (k,l) ) (k,l) (n,m). Since µ (,) = ˆλ (i,j), the proof is complete. Corollary For each C M we have ϕ(c) P and γ(c) = δ(ϕ(c)). In particular, ϕ maps M r,c to P r,c, and ϕ (P r,c ) M r,c. Proof According to Th. 2(ii), relation (2), and Lemmas 3 and 2, the two sequences of partitions that define ϕ(c) satisfy the conditions of Lemma. Thus ϕ(c) is a plane partition. Moreover, δ(ϕ(c)),j = λ (,j) = C (,j) = γ(c),j, and δ(ϕ(c)) i, = γ(c) i, similarly. Thus γ(c) = δ(ϕ(c)). We next prove Th.. The above Corollary establishes that a colored partition with cumulative marginal hook H is mapped by ϕ to a plane partition with diagonal hook H. Bijectivity of ϕ follows from the following Theorem. Theorem 3 The mapping ϕ defined on M by (4) is one-to-one and onto P. Proof Let P be a plane partition, we claim that there exists a unique colored partition C such that P = ϕ(c). For all c > 0, this holds with P P,c trivially: the colored partition C defined by C,j = P,j P,j+ is the only one that satisfies (4). For the same reason, we only consider c > in the sequel. Assume that our claim holds for all plane partitions with r rows, and let P P r+,c for an arbitrary c >. Further let partitions λ (i,) and λ (,j) be defined by { λ (i,) k = P i+k,k for i r + λ (,j). (3) k = P k,j+k for j c Then a sequence (a j ) j c of non-negative numbers and a sequence (λ (2,j) ) j c of partitions may be recursively defined by a j = λ (,j) max(λ (,j+), λ (2,j) ), (4) ˆλ (,j) = λ (,j) a j [], (5) ˆλ (,j) = lgd(λ (2,j+) ; λ (,j+), λ (2,j) ). (6) Indeed, assume that λ (2,j) is well defined this way (recall that λ (2,) is well defined by (3)). Then λ (,j) λ (,j+) Sλ (,j) holds by (3) and Lemma. Moreover, λ (,j) λ (2,j) Sλ (,j) holds either for the same reason if j = or by (6) and Lemmas 3 and 2 if j >. (,j) Thus a j is non-negative. Furthermore, by (4,5) we have ˆλ = max(λ (,j+), λ (2,j) ), hence ˆλ (,j) max(λ (,j+), λ (2,j) ). Since Sˆλ (,j) = Sλ (,j) Lemma 4 applies, hence λ (2,j+) is well defined from (6). Let us complete the first sequence by a c = λ (,c) λ (2,c). (7) 0

11 Next define a matrix P by P i,j = { λ (i j+2,) j λ (2,j i+) if i j i if i j. (8) Since P P we have Sλ (i,) λ (i+,) λ (i,), and Sλ (2,j) λ (2,j+) λ (2,j) follows from (6) by induction as above. Thus P is a plane partition. Moreover, it has r rows by construction, and a number c c of columns. We are now ready to (a) exhibit C with P = ϕ(c), and (b) show its uniqueness. a) Let C be the unique colored partition such that P = ϕ(c ) (recall that C M r,c by Corollary ), and define C M r+,c by C,j = a j, C i,j = C i,j for i 2. (9) By construction, the growth diagram of C (2,) has its first line and column respectively given by λ (i,), 2 i r+, and λ (2,j), j c. According to (7), the shape of the chain C (,c) is λ (,c). Assume that, for a given k such that c > k 0, the shape of C (,c k) is λ (,c k). By Th. 2, the shape of Ĉ (,c k ) is given by lgd(λ (2,c k) ; λ (2,c k ), λ (,c k) ). According to Lemma 4 and (6), this shape is ˆλ (,c k ). Therefore, according to (2) and (5), the shape of C (,c k ) is λ (,c k ). Thus the shape of C (,j) is λ (,j) for all j c. This proves that P = ϕ(c). b) Finally suppose that D M r+,c also satisfies P = ϕ(d). By (4), the first lines and columns of the growth diagrams (λ (i,j) ) and (µ (i,j) ) of C and D are identical. By Lemma 4 and Th. 2 (ii) we thus have ˆλ (,) = max(λ (,2), λ (2,) ) = max(µ (,2), µ (2,) ) = ˆµ (,). Hence C, = D, by (2) and λ (2,2) = µ (2,2) by Lemma 4. Iterating the argument results in C,j = D,j and λ (2,j+) = µ (2,j+) for all j < c. Then C,c = D,c by Th. 2 (i). Therefore, the first lines of C and D are identical, and the first line and column of the growth diagram of C (2,) and D (2,) are identical. Accordingly, ϕ(c (2,) ) = ϕ(d (2,) ), so that C (2,) = D (2,) by hypothesis. Hence the equality of C and D. 5 Sliced permutations Let S n be the set of the permutations of [n] {,..., n}, such a permutation σ being also viewed as a matrix S in M n,n with S i,j = if j = σ(i), else 0. Recall that a descent of σ S n is an integer i [n ] such that σ(i) > σ(i + )}. A sliced permutation of [n] is a triple (σ, α, β) where σ S n, and α (β resp.) is a non-decreasing sequence in {0,..., n } containing all descents of σ (σ resp.). We introduce in this section a bijective correspondence between colored partitions with n parts and sliced permutations of [n]. This correspondence has the property of preserving the growth diagram, in a sense that is made precise in Corollary 3. Besides of its intrinsic combinatorial interest, it proves useful in 6 where our transformation ϕ and the BK correspondence are compared.

12 Sliced permutations are useful when standardizing a colored partition C as described below. First write the 2-line array associated with C with respect to the left lexicographic order (non-increasing abscissas from left to right); then decreasingly relabel the abscissas from C to, and the ordinates by substituting C to the leftmost occurrence of the greatest one, deleting the corresponding point, and iterating while decrementing C. For example, Denote by ɛ(c) the permutation matrix corresponding to this new 2-line array. Each permutation matrix is self-standardized, but infinitely many colored partitions have the same standardization (for instance, empty columns are ignored). We put this right by remembering lines and columns in the colored partition as follows. Let the cutting sequences of C M r,c be defined by cumulating its margins, that is, for k r and l c, k c l r α(c) k = C i,j, β(c) l = C i,j. i= j= Fig. 4 below illustrates the standardization and the cutting sequences of C (,3), for the colored partition C in Fig. 2. In fact, we next show that standardization and cutting sequences form a characterizing sliced permutation. j= i= 2 3 α = (, 4, 7) β = (0, 0, 4) Figure 4: A colored partition and its sliced permutation Theorem 4 The mapping C (ε(c), α(c), β(c)) is a bijection between colored partitions with n parts and sliced permutations of [n]. Moreover, the shapes of C and ɛ(c) are identical. Proof The repeated values and the 0-values in the cutting sequences indicate the empty rows and columns in C. The positive values determine blocks in the matrix ε(c) that record the entries of C, a positive one being turned into a same number of diagonally consecutive -entries. Accordingly, C is uniquely determined by ε(c), α(c), β(c). Denote by l and r the left and right lexicographic orders on N N. From the definition it follows that, given two points (a, b) and (a, b ) in ε(c) that correspond by 2

13 standardization to the points (i, j) and (i, j ) in C, we have a a (i, j) l (i, j ) and b b (i, j) r (i, j ). Since the product order is equal to h v, we have (a, b) (a, b ) if, and only if, (i, j) (i, j ). As a first consequence, standardization provides us with a bijective and length-preserving correspondence between chains in C and ε(c). It follows that they have both the same shape. Moreover, let k be a descent of σ ε(c), and let (a, b) = (k, σ k ) and (a, b ) = (k +, σ k+ ). Keeping the above notations, we have (i, j) l (i, j ) because k < k + and (i, j) > r (i, j ) because σ k > σ k+. Thus i < i, so that k is the i th term in the cutting sequence α(c). This proves that (ε(c), α(c), β(c)) is a sliced permutation. Finally, let (σ, α, β) be a sliced permutation of [n]. There exists a corresponding colored partition C if each block from σ αi +,β j + to σ αi+,β j+ contains 0 s and a sequence of k (k 0) diagonally consecutive s (indeed, let then C i+,j+ = k, with the convention that α 0 = β 0 = 0). Consider such a block and assume it contains two entries σ ij = σ i j = with i > i. If i i > there must be an entry σ i j = with i > i > i unless either i or i is a descent of σ, which is impossible by definition of a sliced permutation. We may thus assume that i = i. Repeating the above argument with respect to the descents of σ, we first obtain j > j, and then j = j. Notice that ε(c) is a colored partition of n 2 with n parts, so that the value of C is not directly conserved. Incidentally, Th. provides us with the following new correspondence. Corollary 2 Permutations of [n] and plane partitions of n 2 with symmetrical diagonal hook, 2,..., n,..., 2, correspond bijectively by ϕ. According to Th. 4, C and ε(c) have identical shapes. More generally, the growth diagrams of C and ε(c) are very close to one another, so that the plane partition ϕ(c) is uniquely determined by the sliced permutation associated with C. Corollary 3 Let (λ) be the growth diagram of ε(c). The growth diagram (λ ) of C is given by λ i,j = λ +αi,+β j, with the convention that α 0 = β 0 = 0. Proof Recall that λ i,j is the shape of C (i,j), thus the shape of ɛ(c (i,j) ) by Th. 4. Deleting or inserting empty rows or columns in a matrix does not affect the shape of the associated multiset of points in L. Accordingly, the shapes of ɛ(c (i,j) ) and ɛ(c) (+α i,+β j ) ) are identical. 6 Solid graph and correspondences Looking at Fig., one guesses that the plane partition obtained from the BK correspondence might be close to the one presented in this paper. The numbers of rows are indeed identical, and the maximal entry in the former indicates the number of columns in the latter. Is it also true that the horizontal margins must be identical? Is there a simple 3

14 algorithm to turn one partition to the other? In fact, we show in this section that these correspondences, as well as the composition of the BK correspondence with transposition, turn a given colored partition into plane partitions with isometric solid graphs. Recall that the solid graph of a plane partition P is the set of points (i, j, k) (N ) 3 that satisfy k P i,j. Conversely, a plane partition is simply a projected representation of its solid graph. Fig. 5 displays three isometric solid graphs and the corresponding plane partitions (two of them already appear in Fig. ) Figure 5: Three isometrical solid graphs. The reverse RSK algorithm turns a multiset of points C, lexicographically ( l ) ordered in a two-line array, into an ordered pair (P, Q) of plane partitions with decreasing columns and same Ferrers shape (the Ferrers shape of a plane partition P is the linear partition λ defined by λ i = max{j : P i,j 0}). The insertion partition P contains the ordinates of the points and the recording partition Q the abscissas. Recall that, in the reverse RSK algorithm, insertion of a value y in a row λ of P is made in place of λ k, where k = min{i : λ i < y} Figure 6: Two-line array and RSK partitions (P, Q). 4

15 The BK transform maps C to the plane partition obtained from the pair (P, Q) by a gluing process that takes advantage of decreasing columns (see, e.g., [8] p.366). Fig. 6 illustrates the two-line array associated with the colored partition of Fig. 2 and the corresponding pair. Gluing this pair results in the leftmost plane partition in Fig. 5, while gluing the reverse pair results in the rightmost one. The latter is the BK transform of the transpose of C (see, e.g., [8]). Let us show how the boundary of the growth diagram of C, thus the plane partition ϕ(c), may be reconstructed from the pair (P, Q). Let P [k] be the plane partition defined from P by substituting 0 to each positive entry less than k. In particular, P [] is P. Theorem 5 For all i, j, the shapes of C (i,) and C (,j) are the Ferrers shapes of Q [i] and P [j] respectively. Proof Let us first show that λ(c (i,) ) is the Ferrers shape of Q [i]. Since points are inserted by decreasing abscissas in the reverse RSK algorithm, Q [i] is, by definition, the recording partition of C (i,). Let ε(c (i,) ) be the permutation matrix obtained by standardizing C (i,) (see 5). Since the two-line array associated with ε(c (i,) ) is the standardization of the one associated with C (i,), the Ferrers shape of Q [i] is the Ferrers shape of the recording partition of ε(c (i,) ). According to a theorem of Greene [4], the latter is λ(ε(c (i,) )), which is also λ(c (i,) ) by Th. 4. Next replace C by its transpose t C. It is known that the recording partition of t C is P, so that λ(( t C) (j,) ) is the Ferrers shape of P [j]. To conclude, recall that ( t C) (j,) = t (C (,j) ) and that λ( t (C (,j) )) = λ(c (,j) ). Denote by S and R, respectively, the symmetry with respect to the plane (i = j) and the rotation of 2π/3 around the axis directed by (,,). If G C denotes the solid graph of ϕ(c) then the solid graph of ϕ( t C) is S(G C ). Th. 5 states that R(G Q ) is the solid graph of the plane partition Q whose j th column is λ(c (,j) ), while R(G P ) is the solid graph of the plane partition P whose i th row is λ(c (i,) ). It follows from an analytic calculation that diagonally gluing P and Q into ϕ(c) precisely amounts to gluing P and Q by the BK device. From a 3D viewpoint, we have the following relations between the solid graph G ϕ of ϕ(c) and the solid graphs G BK and G BKt of the plane partitions associated by the BK correspondence with C and its transpose: G BK = SR(G ϕ ), G BKt = R (G ϕ ). (20) In other words, the latter plane partitions read by rotating the solid graph of ϕ(c) in either direct or indirect sense. This is illustrated in Fig. 5. References [] G. E. Andrews, The Theory of Partitions, Addison-Wesley, 976. [2] T. Britz, S. Fomin, Finite posets and Ferrers shapes. Advances in Mathematics 58 (2000),

16 [3] E. R. Gansner, Acyclic digraphs, Young tableaux and nilpotent matrices, SIAM Journal on Algebraic and Discrete Methods 2 (98), [4] C. Greene, An Extension of Schensted s Theorem, Advances in Mathematics 4 (974), [5] C. Greene, Some partitions associated with a partially ordered set, Journal of Combinatorial Theory A 20 (976), [6] P. A. MacMahon, Combinatorial analysis, Chelsea, New York, 960. [7] R. P. Stanley, The conjugate trace and trace of a plane partition, Journal of Combinatorial Theory 4 (973), [8] R. P. Stanley, Enumerative combinatorics (Vol. 2), Cambridge,