Matrix compositions. Emanuele Munarini. Dipartimento di Matematica Politecnico di Milano

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1 Matrix compositions Emanuele Munarini Dipartimento di Matematica Politecnico di Milano Joint work with Maddalena Poneti and Simone Rinaldi FPSAC 26 San Diego

2 Motivation: L-convex polyominoes A polyomino is a connected finite set of cells in the plane Z Z Some examples are A polyomino is L-convex when every pair of cells can be connected by an internal path with at most one change of direction Using formal power series techniques, in G Castiglione, A Frosini, A Restivo, S Rinaldi, Enumeration of L-convex polyominoes, 25 it was proved that the numbers f n of all L- convex polyominoes with perimeter 2(n + 2) satisfy the recurrence f n+2 = 4f n+ 2f n (n ) Specifically f n =, 2, 7, 24, 42, 2, 2

3 Compositions and 2-compositions A composition of a number n N is any k-tuple (x,, x k ) of positive integers such that x + + x k = n A 2-composition of length k of a number n N is a 2 k matrix [ ] x x M = 2 x k y y 2 y k with nonnegative integer entries, without zero columns, such that the sum of all the entries is n For instance the 2-compositions of n = 2 are: [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2,,,,,, 2 The number of all 2-compositions of n is f n 3

4 Polyominoes and 2-compositions Since the number of all L-convex polyominoes with perimeter 2(n+2) is equal to the number all 2-compositions of n, there exists a bijection between this two combinatorial classes In G Castiglione, A Frosini, E Munarini, A Restivo, S Rinaldi, Enumeration of L-convex polyominoes II Bijection and area, FPSAC 5 it was given an explicit bijection Here follows a sketch 4

5 The bijection a b c a b c a d d a d b b aabcdabcdbdab The factorization in c h a, bd k (h, k ), c r d s (r, s ) implies the bijection: [ ] [ ] [ ] c h h + a, bd k, c r d s r k + s a(c a)(bd )(cd)(c a)(bd )(cd)(bd)(c a)b [ ] 2 5

6 Aim of the talk: to give an elementary introduction to matrix compositions Structure of the talk: definition of matrix compositions recurrences and explicit form Cassini-like identity and combinatorial interpretation some encoding other results 6

7 Definition of m-compositions Let m N, m > An m-row matrix composition (m-composition for short) of length k of a number n N is an m k matrix such that M = x ij N i, j x x k x m x mk (x j,, x kj ) (,, ) j σ(m) = m k i= j= x ij = n C (m) n = { m-compositions of n } C (m) n,k = { m-compositions of n of length k } n = C (m) n n,k = C(m) n,k 7

8 Multisets A multiset on a set X is a function µ : X N The multiplicity of an element x X is µ(x) The order of µ is ord(µ) = x X µ(x) The number of all multisets of order k on a set of size n is the multiset coefficient (( )) n = nk k k! = n(n + ) (n + k ) k! The columns of an m-composition are multisets with positive order on an m-set M = c c m Each column (c,, c k ) is equivalent to the multiset µ : [m] N defined by ( ) 2 m µ = c c 2 c k with positive order: µ() + + µ(m) > 8

9 Sum recursions M = x x 2 x k+ x m x m2 x mk+ n+m k n+m,k+ = i= (( )) m i n+m i,k n+m n+m = i= (( )) m i n+m i 9

10 Main recurrence Let A i be the set of all m-compositions M = x i of n + m with x i Then C (m) n+m = A A m and for the Principle of Inclusion-Exclusion n+m = S [m] S ( ) S A i i S i S A i is formed by all the m-compositions M = x i for every i S Then x i A i = i S 2c(m) n+m S

11 Therefore we have the main recurrence n+m = 2 m i= ( m i ) ( ) i n+m i For m = 2, 3, 4 we have the recurrences: c (2) n+2 = 4c(2) n+ 2c(2) n c (3) n+3 = 6c(3) n+2 6c(3) n+ + 2c(3) n c (4) n+4 = 8c(4) n+3 2c(4) n+2 + 8c(4) n+ 2c(4) n Similarly n+m,k+ = m i= + ( m i ) ( ) i m i= ( m i n+m i,k + ) ( ) i n+m i,k+

12 Explicit form Let A i be the set of all matrices M M m,k (N) where the i-th column is the zero vector and σ(m) = n Then = A A k and for the Principle of Inclusion-Exclusion C (m) n,k n,k = A A k = S [k] ( ) S A i i S An element in i S A i is a matrix M M m,k (N) with a zero vector in each column indexed by the elements of S Removing such columns we just have a multiset of order n on a set of size mk m S Then A i = i S and consequently (( m(k S ) n )) n,k = k i= ( k i ) (( m(k i) n )) ( ) i 2

13 Cassini-like identities The numbers f n = c (2) n of all 2-compositions of n satisfy a Cassini-like identity: for every n f n f n+2 f 2 n+ = 2n Equivalently f n f n+ f n+ f n+2 = 2n For the numbers n such an identity becomes n n+ n+m n+ c(m) n+m n+2 c(m) c(m) n+m c(m) n+m n+2m 2 = ( ) m 2 2 n for every m and n 3

14 First reduction Let C (m) n = n n+ n+m 2 n+m By the main recurrence n+ n+m n+2 n+m c(m) n+m c(m) n+2m 3 n+m c(m) n+2m 2 n+m = α m n+m + + α n+ + α c n (m) where α k = ( ) m k 2 ( m k The last row can be simplified subtracting a suitable linear combination of the first m rows ) 4

15 We obtain the determinant n n+ n+m 2 α n n+ n+m n+2 n+m c(m) n+m c(m) n+2m 3 α n α n+m 2 Extracting α = ( ) m 2 from the last row and shifting cyclically all rows downward we have det C (m) n = 2 det C (m) n Then, for every n, it follows that: where det C (m) n = 2 n det C (m) C (m) = [ i+j+ ]m i,j= To compute the determinant of this matrix it is useful to consider a combinatorial interpretation of m-compositions 5

16 Combinatorial interpretation Given the 3-composition M = consider its entries as the number of occurrences of three given colors, for instance red, green and blue, linearly ordered red < green < blue Then M = is equivalent to a 3-colored bargraph of area with 4 columns 6

17 Now, writing the columns horizontally, from the bottom to the top, we have a 3-colored linear partition of the set {, 2,, } with 4 blocks: In general, the m-compositions of n of length k are equivalent to the m-colored bargraphs of area n with k columns m-colored linear partitions of [n] with k blocks; 7

18 A useful sum Let π be an m-colored linear partition of L = {x,, x i+,, x i+j+ } The element x i+ belongs to a block of the form {x i h+,, x i, x i+, x i+2,, x i+k+2 }: π x i h+ x i+ x i+k+ π 2 i h h k j k Removing this block, π splits into two m-colored linear partitions π and π 2 Then it follows that i+j+ = h,k ( m h + k + ) i h c(m) j k 8

19 Second reduction The previous identity implies the decomposition where C (m) = L (m) M (m) L (m) T L (m) = [ i j ]m i,j= = Then M (m) = [ ( m i + j + c c c c m c c ) ] m det C (m) = det M (m) i,j= To calculate det M (m) we will use another identity coming form our combinatorial interpretation 9

20 Another useful sum Recall that ( ) m i+j+ gives the number of all the order maps f : [i + j + ] [m] Suppose that f(i + ) = k, with k [m] Since f is order preserving, it follows that x i then f(x) k i + 2 x i + j + then k f(x) m Then ( m i + j + ) = = m ( k k= m k= ) ( m k + i j ( i + k) ( m k i j ) ) 2

21 Final step The previous identity implies that where and M (m) = B (m) B (m) B (m) = B (m) = [ ( i + j i [ ( m i Since B (m) = J (m) B (m) where J (m) = [δ i+j,m ] m i,j= = we have and j ) ] m i,j= ) ] m i,j= M (m) = B (m) J (m) B (m), det M (m) = det J (m) (det B (m) ) 2 2

22 Since, as well known, and we have and consequently det J (m) = ( ) m/2 det B (m) = det M (m) = ( ) m/2 det C (m) = det M (m) = ( ) m/2 Finally, since for every n we have then det C (m) n = 2 n det C (m), det C (m) n = ( ) m/2 2 n 22

23 m-compositions as words Consider each m-composition as the concatenation of its columns: M = = Each column (c,, c k ) is equivalent to the multiset µ : [m] N defined by ( ) 2 m µ = c c 2 c k with positive order: µ() + + µ(m) > This means that C (m) = {a µ : µ M (m) } where M (m) is the set of all multisets µ : [m] N with positive order Let X = {x µ : µ M (m) } Then, for a µ x µ, we have the generating series for C (m) c(x) = µ M (m) x µ 3 23

24 In particular, for x µ = x ord(µ) we get (x) = n n x n = h(x) where that is Then h(x) = µ M (m) h(x) = x µ = k (( m ( x) m k )) x k (x) = ( x)m 2( x) m From here we have the previous results and a the following new recurrence n+ = δ n, + 2 n + n k= ( m + k k + ) c (m) n k 24

25 m-compositions of Carlitz type A Carlitz composition is a composition without two equal consecutive elements We say that an m-composition is of Carlitz type when no two adjacent columns are equal For instance M = x 3 3 is of Carlitz type only when x 4 Let Z be the set of all words corresponding to the m-compositions of Carlitz type and let Z µ be the subset of Z formed by the words ending with a µ, for every µ M (m) 25

26 Then Z = + µ M (m) Z µ Z µ = (Z Z µ )a µ µ M (m) To obtain the generating series for Z and Z µ substitute each letter a µ with the indeterminate x µ : z(x) = + µ M (m) z µ (X) = (z(x) z µ (X))x µ Then and finally z µ (X) z µ (X) = x µ + x µ z(x) µ M (m) z(x) = µ M (m) x µ + x µ 26

27 For x µ = x ord(µ) we have n z (m) n x n = k k (( )) m x k + x k where z n (m) is the number of all m-compositions of Carlitz type of n Expanding this series we can obtain: where z (m) n = k α,β N k α β=n (( m α )) ( ) β k α β = a b + + a k b k (( )) m α β = b + + b k = (( )) m a (( )) m a k for every α = (a,, a k ) and β = (b,, b k ) 27

28 A regular language for m-compositions The encoding used in A Björner, R Stanley, An analogue of Young s lattice for compositions, FP- SAC 5 for ordinary compositions can be extended to m-compositions as follows For instance M = = = = = a a a 3 b 2 b a 3 b a a 2 a 3 a 3 28

29 More precisely, given the finite alphabet A m = {a,, a m, b,, b m }, we can define a map l : C (m) A m setting l a, l a m, + l b, + and proceeding as in the example l b m The words of the language L m = l(c (m) ) A m, corresponding to the m-compositions, are characterized by the following conditions: i) the first letter is a or or a m ; ii) each letter a i or b i can be followed by any b j, while it can be followed by a letter a j only when i j 29

30 This characterization implies a unique factorization of the form xy, where: x = a i a i m m ( ε), with i,, i m ; y = y y k (possibly empty), where y r = b j a q j j aq m, q j,, q m Then L m is a regular language defined by the unambiguous regular expression: ε + L ml m where ε is the empty word and L m = a+ a 2 a m + a+ 2 a 3 a m + + a+ m, L m = ( b a a 2 a m + b 2a 2 a m + + b ma m ) This is the basis for an efficient algorithm for the exhaustive generation of m-compositions (Gray code), as described in: E Grazzini, E Munarini, M Poneti, S Rinaldi, On the generation of m- compositions and m-partitions, 26 3

31 Other results: Asymptotic expansion: n n 2m( m 2 ) ( m 2 m 2 ) n m-compositions without zero rows: f (m) n = f (m) (x) = m k= m k= ( m k ( m k ) ( ) n k c (k) n ) ( ) n k ( x) k 2( x) k m-compositions with palindromic rows: n p (m) n x n = ( + x)m 2( x 2 ) m enumeration of various kind of m-colored bargraphs 3