paolo massazza

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On Counting k-Convex Polyominoes(Short Paper)Paolo Massazza11 University of Insubria, ItalyAbstractWe show that, for any fixed integer 𝑘 > 2, the problem of computing the number of 𝑘-convex polyominoesof area 𝑛 is in P.Keywordspolyominoes, counting problem, integer sequence1. IntroductionA polyomino is a geometrical figure consisting of a finite set of connected unit squares (calledcells) in the plane Z× Z, considered up to translations. Polyominoes gained popularity afterthe paper of S. W. Golomb [1]. Nowadays they are widely studied by physicists, mathemati-cians, computer scientists and also by biologists. The problem of counting the number 𝑐𝑛 ofpolyominoes with 𝑛 cells (i.e. of area 𝑛) is probably one of the fundamental open problems incombinatorial geometry (see problem 37 in [2]). Due to the difficulty of the problem, simplerclasses of polyominoes have been introduced. In particular, the class of convex polyominoes(polyominoes where the intersection with an infinite horizontal or vertical stripe is a finitesegment) and some of its subclasses have been investigated [3, 4, 5, 6, 7]. In this paper, weconsider the class Conv𝑘 [8] containing all convex polyominoes 𝑃 with the property that anytwo cells of 𝑃 can be joined by a path in 𝑃 with at most 𝑘 changes of direction, where 𝑘 is afixed integer greater than 2. We show that, for any fixed 𝑘 > 2, the algorithm presented in [9]leads to a set of recurrence equations for computing the number of 𝑘-convex polyominoes ofarea 𝑛 in polynomial time, using 𝑂(𝑛5) space.2. Notation and preliminariesLet 𝑃 be a polyomino with an 𝑟 × 𝑐 minimal bounding rectangle. The rows (resp., columns) of𝑃 are numbered from bottom to top (resp., from left to right). The area A(𝑃 ) of 𝑃 is the numberof its cells. A cell of 𝑃 is identified by a pair of integers (𝑖, 𝑗), where 𝑖 (resp., 𝑗) is the row(resp., column) index. Two cells 𝑎 = (𝑖, 𝑗) and 𝑎′ = (𝑖′, 𝑗′) are adjacent if |𝑖− 𝑖′|+ |𝑗 − 𝑗′| = 1.Given two cells 𝑎 and 𝑏 of 𝑃 , a path in 𝑃 from 𝑎 to 𝑏 is a sequence 𝑞1, 𝑞2, . . . , 𝑞𝑘 of cells of𝑃 , with 𝑞1 = 𝑎 and 𝑞𝑘 = 𝑏, such that 𝑞𝑖 and 𝑞𝑖+1 are adjacent for all 𝑖 with 1 ≤ 𝑖 < 𝑘. AProceedings of the 23rd Italian Conference on Theoretical Computer Science, Rome, Italy, September 7-9, 2022" paolo.massazza@uninsubria.it (P. Massazza)© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).CEURWorkshopProceedingshttp://ceur-ws.orgISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)step is a sequence of two adjacent cells (𝑖, 𝑗), (𝑖′, 𝑗′). Steps are distinguished according to thedirections N (North), W (West), S (South) and E (East). The number of changes of directionin a path 𝛽 ∈ {N,W,S,E}+ is defined as the number of indices 𝑖 such that 𝛽𝑖 ̸= 𝛽𝑖+1, with1 ≤ 𝑖 < |𝛽|. A path is monotone if 𝛽 ∈ {N,W}+ (NW-path) or 𝛽 ∈ {N,E}+ (NE-path) or𝛽 ∈ {S,E}+ (SE-path) or 𝛽 ∈ {S,W}+ (SW-path). A polyomino 𝑃 is horizontally convex (resp.,vertically convex) if any row (resp. column) of 𝑃 consists of one segment. The class of convexpolyominoes contains all polyominoes that are both horizontally and vertically convex. It hasbeen proved [10, Proposition 1] that a polyomino 𝑃 is convex if and only if any two cells of𝑃 are joined by a monotone path in 𝑃 . The degree of convexity of 𝑃 , denoted by deg𝑐(𝑃 ),is defined as the least integer 𝑘 such that any two cells of 𝑃 can be joined by a monotonepath in 𝑃 with at most 𝑘 changes of direction. A convex polyomino is called 𝑘-convex if itsdegree of convexity is at most 𝑘. Given a convex polyomino 𝑃 and its minimal boundingrectangle 𝐵, we say that 𝑃 is a stack (resp., Ferrers diagram, parallelogram, rectangle) if it sharesexactly two adjacent (resp., three, two opposite, four) vertices with 𝐵. A stack 𝑃 is a left (resp.,right) stack if the column with the largest area is the last (resp., first) one. Analogously, in aleft (resp., right) Ferrers diagram the largest column is the last (resp., first) one. We denoteby L (resp., R) the set of left (resp., right) stacks. F𝐿 (resp., F𝑅) is the set of left (resp., right)Ferrers diagrams. Furthermore, we indicate by C (resp., T) the set of parallelograms (resp.,rectangles). For a class A of polyominoes, A(𝑛) is the set of polyominoes in A of area 𝑛. Bylow(𝑗) (resp., high(𝑗)) we denote the row index of the bottom (resp., top) cell of column 𝑗.Similarly, left(𝑖) denotes the column index of the leftmost cell of row 𝑖. Lastly, first(𝑃 )(resp., last(𝑃 )) indicates the first (resp., last) column of 𝑃 . Given two columns 𝑖 and 𝑗, wesay that 𝑖 and 𝑗 are overlapping (resp., disjoint), denoted by 𝑖 ⇄ 𝑗 (resp., 𝑖 ≍ 𝑗), if and only iflow(𝑗) < low(𝑖) ≤ high(𝑗) < high(𝑖) or low(𝑖) < low(𝑗) ≤ high(𝑖) < high(𝑗) (resp.,low(𝑖) > high(𝑗) or low(𝑗) > high(𝑖)). Moreover, we say that 𝑖 includes 𝑗, denoted by 𝑗 ⊆ 𝑖,if and only if low(𝑖) ≤ low(𝑗) and high(𝑖) ≥ high(𝑗). Given a convex polyomino 𝑃 , let 𝑒be the rightmost column of 𝑃 such that 𝑐 ⊆ 𝑒 for 1 ≤ 𝑐 < 𝑒. Then, 𝑃 is called descending(resp., ascending) if there exists a column 𝑗 such that 𝑗 > 𝑒, 𝑗 ⇄ 𝑒 and low(𝑒) > low(𝑗) (resp.,low(𝑒) < low(𝑗)). The set of descending (resp., ascending) 𝑘-convex polyominoes is indicatedby DConv𝑘 (resp., AConv𝑘). The set of all descening polyominoes is DConv. If 𝑃 is neitherdescending nor ascending then 𝑃 ∈ T∪F𝐿∪F𝑅∪L∪R or it belongs to the class LR containingall convex polyominoes that are the concatenation of two polyominoes, 𝑃 = 𝑃1 · 𝑃2, where𝑃1 ∈ L ∪ F𝐿, 𝑃2 ∈ T ∪ R ∪ F𝑅 and first(𝑃2) ⊊ last(𝑃1). Notice that any 𝑃 ∈ LR containsa column ?̄? such that 𝑗 ⊆ ?̄? for all columns 𝑗, hence deg𝑐(𝑃 ) ≤ 2. Lastly, a cell (𝑖, 𝑗) of 𝑃 isa NW-corner if (𝑖, 𝑗 − 1) and (𝑖+ 1, 𝑗) are not cells of 𝑃 . Analogously, we define NE-corners,SE-corners and SW-corners. Two corners are called opposite if either one is a NW-corner andthe other is a SE-corner, or one is a NE-corner and the other is a SW-corner. Clearly, onehas Conv𝑘 = T ∪ F𝐿 ∪ F𝑅 ∪ L ∪ R ∪ LR ∪ AConv𝑘 ∪ DConv𝑘 and (because of symmetry)|DConv𝑘(𝑛)| = |AConv𝑘(𝑛)|, |L(𝑛)| = |R(𝑛)|, |F𝐿(𝑛)| = |F𝑅(𝑛)|. Since all unions are disjoint,it follows that|Conv𝑘(𝑛)| = |T(𝑛)|+ 2 · |F𝐿(𝑛)|+ 2 · |L(𝑛)|+ |LR(𝑛)|+ 2 · |DConv𝑘(𝑛)|. (1)Thus, the counting problem for Conv𝑘 is reduced to computing |DConv𝑘(𝑛)| and to someother counting problems that are immediately solved in polynomial time. In particular, we can30123201233Figure 1: The arrows show the first steps of the paths determined by [9, Lemma 8] and associated witha SW corner. The integer below a column indicates its degree of convexity.compute |LR(𝑛)| in polynomial time as shown in [11]. In order to compute |DConv𝑘(𝑛)| wedefine a decomposition for descending polyominoes.Definition 2.1 (standard decomposition). A polyomino 𝑃 ∈ DConv can be decomposed as𝑃 = 𝐿 ·𝐹 ·𝐶 ·𝑅 (with 𝐹 , 𝑅 possibly empty) for suitable polyominoes 𝐿 ∈ L∪T∪F𝐿, 𝐹 ∈ F𝑅,𝐶 ∈ C ∪ T ∪ F𝑅, and 𝑅 ∈ R ∪ T ∪ F𝑅 such that: first(𝐹 ) ⊊ last(𝐿), low(last(𝐿)) =low(first(𝐹 )), last(𝐹 ) ⇄ first(𝐶) (or last(𝐿) ⇄ first(𝐶) if 𝐹 = 𝜖), low(last(𝐿)) >low(first(𝐶)) and (if 𝑅 ̸= 𝜖) first(𝑅) ⊊ last(𝐶), low(last(𝐶)) < low(first(𝑅)).We stress that the standard decomposition of 𝑃 is unique (e.g. last(𝐿) is the rightmostcolumn ?̄? of 𝑃 such that 𝑗 ⊆ ?̄? for 𝑗 < ?̄?). The subset of DConv2 containing polyominoesdecomposed as 𝐿 ·𝐶 ·𝑅 (resp., 𝐿 ·𝐶 , 𝐿 ·𝐹 ·𝐶 ·𝑅, 𝐿 ·𝐹 ·𝐶) is LCR2 (resp., LC2, LFCR2, LFC2).Thus, for any 𝑘 > 2 one has the partition DConv𝑘 = LFCR𝑘 ∪ LFC𝑘 ∪ LCR𝑘 ∪ LC𝑘.By [9, Thm. 6], the degree of convexity of 𝑃 ∈ DConv depends on the number 𝑘 of changesof direction in a particular NW-path that starts at a SE-corner 𝑎 and ends at a NW-corner 𝑏. Thisis a path where the first 𝑘− 1 changes of direction always occur on the boundary of 𝑃 , whereasthe 𝑘th one possibly occurs on a cell that is not on the boundary. Furthermore, from [9, Thm. 6]it follows that the degree of convexity of the 𝑗th column of 𝑃 is deg𝑐(𝑃, 𝑗) = max{𝐷(𝑎, 𝑏)|𝑎 =(low(𝑗) , 𝑗), 𝑏 is a NW-corner of 𝑃}, where 𝐷(𝑎, 𝑏), is the least integer k such that there existsa monotone path in 𝑃 from 𝑎 to 𝑏 with k changes of direction.Given 𝑃 = 𝐿 · 𝐹 · 𝐶 · 𝑅 and a column 𝑗 in 𝐹 · 𝐶 · 𝑅, we know from [9, Lemma 8]that deg𝑐(𝑃, 𝑗) = min(deg𝑐(𝑃, 𝑗′) + 2,deg𝑐(𝑃, 𝑗′′) + 1), where 𝑗′ = left(high(𝑗)) and𝑗′′ = left(low(𝑗)). see Fig. 1 for an example. In the sequel, we consider each column 𝑐 of 𝑃as the concatenation of vertical segments associated with different degrees of convexity. Thesesegments are identified by considering the degrees of convexity of the leftmost columns thatare reached by W-paths starting from cells of 𝑐.Definition 2.2 (𝛽-segment). A cell (𝑖, 𝑗) of 𝑃 ∈ DConv belongs to a 𝛽-segment if and only ifeither 𝛽 > 0 ∧ 𝑖 < low(first(𝑃 )) and the cell (𝑖, left(𝑖)) belongs to a column of degree ofconvexity 𝛽, or 𝛽 = 0 ∧ 𝑖 ≥ low(first(𝑃 )).From [9, Lemma 8], if deg𝑐(𝑃, 𝑗) = 𝛼 then column 𝑗 may contain only 𝛽-segments with𝛼 − 2 ≤ 𝛽 ≤ 𝛼. Notice that (𝑖, 𝑗) belongs to a 𝛽-segment if and only if (𝑖, 𝑗 − 1) belongsto a 𝛽-segment or (𝑖, 𝑗 − 1) is not in 𝑃 and deg𝑐(𝑃, 𝑗) = 𝛽. Fig. 1 shows how the columnsof a polyomino in DConv consist of at most three 𝛽-segments (depicted as vertical segmentscomprising cells of the same color).Figure 2: A polyomino counted by 𝐶(16, 5, 3, 3, 1, 2).3. CountingWe focus on computing |DConv𝑘(𝑛)| since the remaining values in (1) are easily computedin polynomial time (see, for instance, [11]). Given 𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖 ∈ N, with 𝑛, 𝑏 > 0 and𝑥+𝑦 ≤ 𝑏, let 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖) be the number of polyominoes 𝑃 in LFC𝑘(𝑛)∪LC𝑘(𝑛) such thatA(last(𝑃 )) = 𝑏, deg𝑐(𝑃, last(𝑃 )) = 𝑗, low(last(𝑃 )− 1)− low(last(𝑃 )) = 𝑖, and wherelast(𝑃 ) contains a 𝑗-segment of area 𝑥, a (𝑗 − 1)-segment of area 𝑦 and a (𝑗 − 2)-segment ofarea 𝑏−𝑥−𝑦, see Fig. 2. Obviously, one has |LFC𝑘(𝑛)|+|LC𝑘(𝑛)| =∑︀𝑏,𝑗,𝑥,𝑦,𝑖𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖),and the problem boils down to compute 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖) efficiently.Consider the standard decomposition of a polyomino 𝑃 counted by 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖), 𝑃 =𝐿 · 𝐹 · 𝐶 or 𝑃 = 𝐿 · 𝐶 . If the second to last column of 𝑃 belongs to 𝐶 then 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖)depends on 𝐶(𝑛 − 𝑏, 𝑏′, 𝑗′, 𝑥′, 𝑦′, 𝑖′) for suitable 𝑏′, 𝑗′, 𝑥′, 𝑦′, 𝑖′. Otherwise, the second to lastcolumn of 𝑃 is in 𝐹 or in 𝐿, and 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖) is computed using the values 𝐿(𝑚′, 𝑝′, 𝑞′, 𝑦′)and 𝐹 (𝑚′, 𝑝′, 𝑞′), where 𝐿(𝑚′, 𝑝′, 𝑞′, 𝑦′) (resp., 𝐹 (𝑚′, 𝑝′, 𝑞′)) is the number of 𝑃 ∈ L(𝑚′) ∪F𝐿(𝑚′) ∪ T(𝑚′) (resp. 𝑃 ∈ F𝑅(𝑚′)) such that A(first(𝑃 )) = 𝑝′, A(last(𝑃 )) = 𝑞′ andlow(first(𝑃 ))−low(last(𝑃 )) = 𝑦′. Here we consider only the case when column last(𝑃 )−1 is in 𝐶 (we refer to the full paper for the remaining cases). This is the most complex caseas it leads to four different equations, depending on the conditions 𝑥 > 𝑖 (1), 𝑥 = 𝑖 ∧ 𝑦 = 0(2), 𝑥 = 𝑖 ∧ 𝑦 > 0 ∧ 𝑖 > 0 (3) and 𝑥 = 𝑖 ∧ 𝑦 > 0 ∧ 𝑖 = 0 (4). In case (1), column last(𝑃 )− 1always has degree of convexity 𝑗, since the relation 𝑥 > 𝑖 implies the existence of a columnof degree of convexity 𝑗 to the left of last(𝑃 ), and the sequence of degrees of convexity ofcolumns in 𝐶 is not decreasing by [9, Thm. 9]. Furthermore, one necessarily has 𝑥 + 𝑦 < 𝑏because last(𝑃 ) contains a (𝑗 − 2)-segment, see Fig. 3. So, it follows that 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖) =∑︀𝑛−𝑏−2𝑏′=𝑏−𝑖∑︀𝑥−𝑖𝑖′=0𝐶(𝑛 − 𝑏, 𝑏′, 𝑗, 𝑥 − 𝑖, 𝑦, 𝑖′). In case (3) (see Fig. 4) the recurrence equation is𝐶(𝑛, 𝑏, 𝑗, 𝑖, 𝑦, 𝑖) =∑︀𝑛−𝑏−2𝑏′=𝑏−𝑖+1∑︀𝑏′−𝑏+𝑖−1𝑦′=𝑏−𝑖−𝑦∑︀𝑦𝑖′=0𝐶(𝑛− 𝑏, 𝑏′, 𝑗 − 1, 𝑦, 𝑦′, 𝑖′) +∑︀𝑛−𝑏−2𝑏′=𝑏−𝑖 𝐶(𝑛−𝑏, 𝑏′, 𝑗, 0, 𝑦, 0). In fact, the second to last column of a polyomino 𝑃 counted by 𝐶(𝑛, 𝑏, 𝑗, 𝑖, 𝑦, 𝑖)may have degree of convexity 𝑗−1 or 𝑗, but not 𝑗−2. Indeed, if one had deg𝑐(𝑃, last(𝑃 )−1) =𝑗−2 then the sequence of degrees of convexity in𝐶 would be decreasing, since last(𝑃 ) containsa (𝑗 − 1)-segment and this implies the existence of a column 𝑚 to the left of last(𝑃 ) suchthat deg𝑐(𝑃,𝑚) = 𝑗 − 1. Furthermore, 𝑥 = 𝑖 ∧ 𝑖 > 0 implies that last(𝑃 ) contains a (𝑗 − 2)-segment. As a consequence the (𝑗− 1)-segments in last(𝑃 )− 1 and in last(𝑃 ) have the samearea. So, if deg𝑐(𝑃, last(𝑃 )− 1) = 𝑗 − 1 we sum the values 𝐶(𝑛− 𝑏, 𝑏′, 𝑗 − 1, 𝑦, 𝑦′, 𝑖′) on all𝑏′, 𝑦′, 𝑖′ such that:Figure 3: Possible configurations when 𝑥 > 𝑖.Figure 4: Possible configurations when 𝑥 = 𝑖 and 𝑥, 𝑦 > 0.• the area 𝑏′ of column last(𝑃 )− 1 is at least 𝑏− 𝑖+ 1 (since 𝑏′ must contain a (𝑗 − 3)-segment) and at most 𝑛− 𝑏− 2 (since A(𝐿) ≥ 2);• the area 𝑦′ of the (𝑗 − 2)-segment in last(𝑃 ) − 1 is at least 𝑏 − 𝑖 − 𝑦 (the area of the(𝑗 − 2)-segment in last(𝑃 )) and at most 𝑏′ − 𝑏 + 𝑖 − 1 (because column last(𝑃 ) − 1must contain a (𝑗 − 3)-segment);• 𝑖′ is at least 0 and at most 𝑦 (the lower 𝑖′ cells of last(𝑃 ) − 1 has degree 𝑗 − 1 sinceleft(low(last(𝑃 )− 1) + 𝑒) = last(𝑃 )− 1 for 0 ≤ 𝑒 < 𝑖′).Otherwise, if deg𝑐(𝑃, last(𝑃 )− 1) = 𝑗 we sum the values 𝐶(𝑛− 𝑏, 𝑏′, 𝑗, 0, 𝑦, 0) on all 𝑏′ nosmaller than 𝑏− 𝑖 and no greater than 𝑛− 𝑏− 2. From the existence of a (𝑗 − 1)-segment inlast(𝑃 ) and deg𝑐(𝑃, last(𝑃 )−1) = 𝑗, it follows low(last(𝑃 )− 2)−low(last(𝑃 )− 1) = 0.Hence, last(𝑃 )− 1 cannot contain a 𝑗-segment. The recurrence equations for cases (2) and(4), as well as for the cases associated with the set LCR𝑘 ∪ LFCR𝑘, are obtained by a similarreasoning.Based on these equations, we have developed a C++ program to compute |Conv𝑘(𝑛)| for𝑘 > 2 and 𝑛 > 0. This program has space complexity 𝑂(𝑛5) (coming from the size of the tablecontaining the values 𝐶(𝑛, 𝑏, 𝑗, 𝑥, 𝑦, 𝑖)) and produced Table 1 in few minutes (on a MacbookPro). We point out that this integer sequence does not appear in OEIS.References[1] S. W. Golomb, Checker boards and polyominoes, Amer. Math. Monthly 61 (1954) 675–682.[2] E. D. Demaine, J. S. B. Mitchell, J. O’Rourke, The open problems project, last update 2020.URL: http://cs.smith.edu/~jorourke/TOPP.Table 1The number of 3-convex polyominoes of area 𝑛 for 0 ≤ 𝑛 ≤ 40.0, 1, 2, 6, 19, 59, 172, 470, 1206, 2934, 6812, 15192, 32709, 68282, 138678, 274822, 532719, 10121441888226, 3464168, 6258249, 11146013, 19590450, 34011064, 58371083, 99103808, 166563604,277281796, 457451501, 748274488, 1214117566, 1954879052, 3124637754, 4959621329,7819943680, 12251575214, 19077932142, 29534613958, 45466767846, 69616951878, 106043316448[3] M. Bousquet-Mélou, Convex polyominoes and heaps of segments, Journal of Physics A:Mathematical and General 25 (1992) 1925–1934. URL: https://doi.org/10.1088/0305-4470/25/7/031. doi:10.1088/0305-4470/25/7/031.[4] E. Barcucci, R. Pinzani, R. Sprugnoli, Directed column-convex polyominoes by recurrencerelations, in: M. C. Gaudel, J. P. Jouannaud (Eds.), TAPSOFT’93: Theory and Practice ofSoftware Development, Springer Berlin Heidelberg, Berlin, Heidelberg, 1993, pp. 282–298.[5] M. Bousquet-Mélou, A method for the enumeration of various classes of column-convexpolygons., Discrete Math. 154 (1996) 1–25.[6] A. Del Lungo, M. Nivat, R. Pinzani, S. Rinaldi, A bijection for the total area of parallelogrampolyominoes, Discret. Appl. Math. 144 (2004) 291–302. URL: https://doi.org/10.1016/j.dam.2003.11.007. doi:10.1016/j.dam.2003.11.007.[7] G. Castiglione, A. Restivo, Ordering and convex polyominoes, in: MCU 2004, volume 3354of Lecture Notes in Comput. Sci., Springer, 2005, pp. 128–139.[8] A. Micheli, D. Rossin, Counting k-convex polyominoes, Electron. J. Comb. 20 (2013).[9] S. Brocchi, G. Castiglione, P. Massazza, On the exhaustive generation of k-convex poly-ominoes, Theor. Comput. Sci. 664 (2017) 54–66.[10] G. Castiglione, A. Restivo, Reconstruction of l-convex polyominoes, Electronic Notes in Dis-crete Mathematics 12 (2003) 290 – 301. URL: http://www.sciencedirect.com/science/article/pii/S1571065304004949. doi:https://doi.org/10.1016/S1571-0653(04)00494-9,9th International Workshop on Combinatorial Image Analysis.[11] V. Dorigatti, P. Massazza, On counting l-convex polyominoes, in: 22nd Italian Conferenceon Theoretical Computer Science, 2021, volume 3072 of CEUR Proceedings, 2021, pp. 193–198.

On Counting k-Convex Polyominoes

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On Counting k-Convex Polyominoes

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