title a note on tatami tilings (mathematical foundation of ... · nextit.to since the height is...

Title A Note on Tatami Tilings (Mathematical Foundation ofAlgorithms and Computer Science)

Author(s) ALHAZOV, Artiom; MORITA, Kenichi; IWAMOTO, Chuzo

Citation 数理解析研究所講究録 (2010), 1691: 1-7

Issue Date 2010-06

URL http://hdl.handle.net/2433/141580

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

A Note on Tatami Tilings

Artiom ALHAZOV1’2*, Kenichi MORITA1, Chuzo IWAMOTO11 IEC, Department of Information Engineering

Graduate School of Engineering, Hiroshima UniversityHigashi-Hiroshima 739-8527 Japan

E-mail: {morita@iec.,chuzo@}hiroshima-u.ac.jp2 Institute of Mathematics and Computer Science

Academy of Sciences of MoldovaAcademiei 5, Chi\c{s}in\u\{a}u MD-2028 Moldova

E-mail: [email protected]

Abstract Inspired by the rules ofJapanese tatami layouts, we present inthese paper some characterization resultsof tiling rectangles with lx2 and $2x1$ blocksin case a single lxl block is allowed. Thetatami condition is that four blocks cannotmeet in the same point.

1 Introduction

In Japan, the size of a room is typically mea-sured by the number of tatami mats (jo).There are various rules concerning the num-ber and layout of tatami mats. In this pa-per we focus on just one rule: in any layoutthere is never a point where the corners ofthree or four mats touch. It is equivalent tothe requirement that four mats never meetin one point.

$\overline{*ArtiomAlha\prime z}$ov gratefully acknowleges the sup-port of the Japan Society for the Promotion of Sci-ence and the Grant-in-Aid for Scientific Research,project $20\cdot 08364$ . He also acknowledges the supportby the Science and Technology Center in Ukraine,project 4032.

$\square \square \square ^{\Pi^{\square }}\square$翫 $\coprod_{\square ^{\square }\square }\square \square \square \prod_{\square \square }$

Figure 1: $6- j\overline{o}$ and 4.5-j6layouts

We formalize this concept in Section 2and we call tilings satisfying this propertythe tatami tilings. The case when onlydominoes are used has been characterizedin [1] and [2]; one explains why the numberof tilings is unexpectedly small when the ra-tio of dimensions is not big. In Subsection3.1 we recall the known results for tatamitilings. In Subsection 3.2 we characterizetilings of odd area, using dominoes and asingle monomino. In Section 4 we prove thischaracterization.

2 DefinitionsWe formalize a rectangular array as a set$A=\{(i, j)|1\leq i\leq m, 1\leq j\leq n\}$ ,and call $m$ and $n$ its height and width, re-spectively. We recall a classic algebra fact

数理解析研究所講究録第 1691巻 2010年 1-7 1

A. Alhazov, K. Morita, C. Iwamoto A Note on Tatami Tilings

that an equivalence relation defined on a setinduces a partition on it, and vice versa. Adomino tile (if there is no confusion, we mayomit the word domino) is a subset of $A$ oftype $\{(i,j), (i+1,j)\}$ or $\{(i,j), (i,j+1)\}$ ;we call it horizontal in the first case and ver-tical in the second case. A monomino is asingle element of $A$ . A tiling is a partitionof a rectangular grid in tiles.

A tiling $T$ is called a tatami tiling if forevery $c,$ $d,$ $1\leq c\leq m-1,1\leq d\leq n-1$ ,elements $(c, d),$ $(c+1, d),$ $(c, d+1),$ $(c+1,$ $d+$

1 $)$ belong to at most 3 tiles of $T$ .Figure 1 graphically represents tatami

tilings with height 3 and widths 4 and 3.

3 Building blocks

3.1 Even area tilings

In this subsection we speak of tatami tilingsthat only use domino tiles. Naturally, thearea of the corresponding rectangles mustbe even (whereas if a single monomino isused, then the area is odd). Without re-stricting generality, assume that $m\leq n$ . Ithas been shown in [1] (see also [2]) that alltatami tilings are horizontal compositions ofspecific building blocks. For even $m$ , tilingsconsist of $H_{m}$ and $V_{m}$ with blocks $L_{m}$ be-tween them and, possibly, on the left andright edges, see Figure 2.

Figure 2: Tilings $H_{10},$ $V_{10}$ and $L_{10}$

For odd $m$ , tilings consist of $H_{m},$ $V_{m}$ andtheir vertical reflections, see Figure 3.

Figure 3: Tilings $H_{9}$ and $V_{9}$

In [1] one defines the function $t(m, n)$

of the number of tatami tilings $m$ by $n$

and describes how to compute it. In [2]one also gives the corresponding generatingfunctions.

3.2 Odd area tilings

Without restricting generality we assume$m\leq n$ . We claim that in case a singlemonomino is used, all tatami tilings are hor-izontal compositions of the building blocks(and reflections of some of them) $H_{m},$ $V_{m}$ ,$M_{m},$ $CH_{m},$ $CV_{m},$ $L_{m}$ , with the followingconditions: only one of the latter four blocksis used in a tiling, and exactly once; block$CH_{m}$ or $CV_{m}$ or its horizontal reflection canonly be used in the left (only in the right ifit is also vertically reflected). The blocksare illustrated in Figure 4, and the proofsfollow after that.

$\Vert\square$

Figure 4: Tilings $H_{9},$ $V_{9},$ $M_{9},$ $CH_{9},$ $CV_{9}$

and $L_{9}$

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4 Auxiliary results forproofs

Lemma 1 Let $A=\{(i, j)$ $|$ $1$ $\leq i\leq$

$m$ , 1 $\leq j\leq n\}$ and $B\subseteq$ $A$ is a re-gion of a rectangle. Let $B_{1}=\{(i, j)\in$

$B|i+j$ is even} and $B_{2}=\{(i,j)\in B|$

$i+j$ is odd}. Then $B$ can only be tiled if$|B_{1}|-|B_{2}|$ is either $-1,0$ or 1, and themonomino is in $B_{2}$ , absent or in $B_{1}$ , re-spectively.

Proof. $B_{1}$ and $B_{2}$ represent a “checker-board” partition of $B$ , and any domino con-sists of one element of $B_{1}$ and one elementof $B_{2}$ . $\square$

Corollary 1 If the area of a rectangle isodd, then one monomino $(i, j)$ is used where$i+j$ is even.

Proof. By Lemma 1, taking $B=A$ . $\square$

Corollary 2 A tnangular corner (see Fig-ure 5) cannot be tiled, except if its area is 1or 3.

Figure 5: Triangle

Proof. If the height of a triangle $B$ is $m$

by $m$ , then its area is $m(m+1)/2$ . Partitionit as in Lemma 1, then

$||B_{1}|-|B_{2}||\square =$

$\lfloor(m+1)/2\rfloor$ .

Corollary 3 A right trapezoid region of arectangle (see Figure 6) cannot be tiled bydominoes only, unless its height is 1.

Figure 6: Right trapezoid

Proof. If the height of the trapezoid is$m$ , then partitioning it as in Lemma 1 yields$||B_{1}|-|B_{2}||$ equal to either $\lfloor(m+1)/2\rfloor$

or (if it is a union of a triangle with height$m$ and an odd-area rectangle) $\lfloor(m-1)/2\rfloor$ .The statement follows from Lemma 1. $\square$

For the next result we join two righttrapezoids with equal long bases, see Fig-ure 7, left.

Lemma 2 A $t$ pentagon “ region (see Figure7, left) cannot be tiled by dominoes only, un-less one of two leftmost elements is in theme ddle row).

Figure 7: Pentagon region from Lemma 2

Proof. Such a pentagon $B$ consists ofa symmetrical region and a smaller righttrapezoid $T$ , see Figure 7, right. Notice thatin the symmetrical region the number of el-ements the sum of whose coordinates is oddis equal to the number of elements the sumof whose coordinates is even, so the valueof $||B_{1}|-|B_{2}||$ in the partitioning definedin Lemma 1 is the same as the correspond-ing value for the trapezoid $T$ . By Corollary3, this value cannot be $0$ unless the height

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of $T$ is 1, and the statement of this lemma Proof. (idea) The claim follows byfollows from Lemma 1. $\square$ a proof technique similar to the one from

Theorem 2.2 of [2], by proceeding as if themonomino was a part of horizontal domino

Lemma 3 If a monomino does not touch extending beyond the left edge of the rect-any side of a rectangle $m$ by $n,$ $m\leq n$ , then angle. The only special case is row 2 or rowit must be in row $(m+1)/2$ . $m-1$ (excluded by Corollary 1). $\square$

Figure 8: Tiling around a monomino

Proof. The claim trivially holds for $m=1$ ,so from now on we suppose $m\geq 3$ . Theremust be at least one domino touching themonomino with its long side, otherwise thetatami condition would not hold. To satisfythe tatami condition, there must be anotherdomino touching both the monomino andthe domino with its long side. Similarly, athird domino must touch the monomino andthe second domino with its long side, anda fourth domino must touch the monominoand the third domino with its long side.This creates a 3 by 3 block. While the blockdoes not touch any side of the rectangle, bysimilar logic it is clear that it is a part of alarger square block with odd dimensions.

Multiple tiles are forced, see Figure 8.Disregarding the vertical tiles in the rightpart, it is clear that a pentagon region (seedashed lines) is isolated. The lemma state-ment follows from Lemma 2. $\square$

Lemma 4 A monomino cannot touch onlythe left side of an odd-area rectangle $m$ by$n,$ $m\leq n$ .

Lemma 5 A horizontal domino cannottouch the left side of an odd-area rectan-gle $m$ by $n,$ $m\leq n$ , unless it is in rows1, 2, $m-1,$ $m$ .

Proof. Suppose a horizontal dominotouches the left side of the rectangle only.The width is odd, so consider the leftmostnon-horizontal tatami $t$ in this row. If $t$ isa monomino and it touches the right side ofthe rectangle, then the claim is reduced toLemma 4. If $t$ is a monomino and it does nottouch the right side of the rectangle, thentwo horizontal tiles are forced above and be-low the left edge of $t$ , and any tatami to theright of $t$ contradicts the tatami condition.

If $t$ is a vertical domino, then the proofis analogous to the proof of case 2 in The-orem 2.2 of [2], with one difference in sub-case 2-1, namely, if the top left triangle issmall. The top left triangle of 1 block isexcluded from the claim of the lemma, sothe remaining case is when the top left tri-angle consists of 3 blocks. Notice that thistriangle is isolated, so we can freely chooseto put a monomino in the leftmost column.However, Lemma 4 shows that such tiling isimpossible, concluding the proof. $\square$

5 Proving the tilingcharacterization

Without restricting generality, we again as-sume $m\leq n$ . It follows from Lemma 4 and

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Lemma 5 that the left edge of the rectangleis touched by vertical dominoes only, excepta monomino may be placed in one cornerand horizontal dominoes may be placed inthe top two rows and the bottom two rows.Hence, only the following cases are possible.

$\bullet$ No horizontal tiles touch the left edge.Since the height is odd, the monominomust be placed in the corner, so we ob-tain block $L_{m}$ , see Figure 4.

$\bullet$ A horizontal tile touches the second toprow or the second bottom row. Dueto Lemma 5, the monomino must beplaced in the corner next to it. Sincethe height is odd, another horizontaltile must be placed in the other left cor-ner, and the rest of the left column istiled by vertical dominoes. More ver-tical tiles are forced, and then multiplehorizontal tiles are forced, see Figure 9,right. Then we have two choices how totile the two leftmost empty elements.Using a vertical domino forces us tocomplete $CV_{m}$ block, see Figure 4. Us-ing two horizontal dominoes forces usto complete $CH_{m}$ block, see Figure 4.

$\bullet$ Neither of the previous cases hold.Since the height is odd, exactly one hor-izontal tile must be placed in one ofthe left corners, and the rest of the leftcolumn is tiled by vertical dominoes.More vertical tiles are forced, and thenwe have choices.

-We do not use the monomino inthe top or bottom row. We recallthat in this case the monominocannot be used except in the mid-dle row. Hence, multiple horizon-tal tiles are forced, see Figure 9,left. Then we have three choiceshow to tile the two leftmost emptyelements. Using the monomino in

the middle row forces us to com-plete $M_{m}$ block, see Figure 4. Us-ing a vertical domino forces usto complete $V_{m}$ block, see Figure4. Using two horizontal dominoesforces us to complete $H_{m}$ block,see Figure 4.

-The monomino is in the top orbottom row. The top row isexcluded by Corollary 1. Themonomino in the bottom rowgives us the following cases: 1)wecomplete $CV_{m}$ or $CH_{m}$ rotatedby 180 degrees, see Figure 4, $2$ ) acontradiction caused by some hor-izontal domino $t$ and two otherdominoes over and under the rightpart of $t$ with common cornerswith $t$ (see Figure 10), so thetiling cannot continue to the right,while another horizontal dominois forced to the right, excludingthe possibility that $t$ touches theright side of the rectangle.

$\coprod\coprod_{\square \square }\coprod_{\square ^{\coprod_{\square }\coprod^{\square }}}\coprod^{\square \square \square }\square \coprod_{\square }\square \square \square _{\square }$ $\square \square \square \square _{\square }^{\coprod^{\square _{\square }\square }}\square \coprod_{\coprod_{\square ^{\square _{\square }}}}^{\square \square \square \square }$

$\square \square \square \square \square \square \square$

Figure 9: Tiling left-to right

$\prod_{\square ,\square }$

$\square \coprod_{\square }$

Figure 10: Tiling cannot continue to theright

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In each case, one of the 6 mentionedblocks is completed, and then we proceedwith the remaining rectangle. Since themonomino has to be used exactly once, anytiling will have exactly one occurrence of$L_{m},$ $M_{m},$ $CH_{m}$ or $CV_{m}$ . It is also easy tosee that if a monomino is used in a cornerof a block, then this must be a corner of en-tire rectangle: any possible left neighbor ofthe horizontal domino under the monominowould violate the tatami condition.

6 Enumerating thetilings

We recall that the function $t(m, n)$ was de-fined in [1], see also [2]. Similarly, we de-fine the function $t_{1}(m, n)$ of tilings whereat most one monomino is used. Note thatreflections and rotations of the same tilingare counted multiple times, unless thattiling has corresponding symmetries. Since$t_{1}(m, n)=t(m, n)$ if $mn$ is even, it suf-fices to define $t_{1}$ for odd values of $m$ and$n$ ( $t(m,$ $n)$ was $0$ there). Clearly, $t_{1}(m, n)=$

$t_{1}(n, m)$ , so it suffices to considcr the casc$m\leq n$ .

It is very easy to see that in case of onlyone row and $n$ columns, the tiling is definedby the position of the monomino between$(n-1)/2$ dominoes, so

$t_{1}(1, n)=(n+1)/2$ .

Finally, if $m\geq 3$ , then we split the functionas follows:

$t_{1}(m, n)=t_{2}(m, n)+t_{3}(m, n)$ ,

where $t_{2}$ enumerates the tilings with$CH_{m}/CV_{m}$ and $t_{3}$ enumerates all othertilings. We also need the function $c(m, n)$

of the number of representations of $n$ as anordered sum of numbers $m+1$ and $m-1$

(we only need it for odd values of $m\geq 3$ ):

$c(m, n)=\{\begin{array}{ll}0 n<01 , n=0,c(m, n-m+1) +c(m, n-m-1) , n>0.\end{array}$

If $m,$ $n$ are odd and $m\geq 5$ , then

$t_{2}(m, n)=4c(m, n-m)+4c(m, n-m-2)$ .

Indeed, every tiling with $CH_{m}/CV_{m}$ con-sists of a tiling by dominoes only, obtainedby removing that block. Multiplier 4 is usedto account for the reflections. Note that theblock $CV_{3}$ degenerates into $L_{3}$ , and the cor-responding tilings are counted in $t_{3}$ . There-fore, $t_{2}(3, n)=4c(m, n-m)$ .

We also assume $t_{3}(m, n)=0$ if $n\leq 0$ . If$m,$ $n$ are odd and $m\geq 3$ , then

$t_{3}(m, n)=2c(m, n-1)+2c(m, n-m)+$$t_{3}(m, n-m-1)+t_{3}(m, n-m+1)$ .

Indeed, the above formula sums up the casesdepending on whether the leftmost block is$L_{m},$ $M_{m},$ $H_{m}$ or $V_{m}$ , respectively. Multiplier2 is used to account for vertical reflections.

Counting only incongruent tilings couldbe done in a similar way, but we do not focuson this here. Below is a table of $t_{1}$ for smallvalues of $m$ and $n$ .

7 Generating functionsAn ordinary generating function of a se-quence is a formal power series whose co-efficients coincide with elements of that

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sequence. The generating functions forthe number of domino tilings were givenin [2]. We now focus on $t’(m, n)$ $=$

$t_{1}(m, n)-t(m, n)$ , i.e., on the number oftilings using exactly one monomino. As-sume $m$ is fixed and consider the sequence$t’(m, m),$ $t’(m, m+1),$ $\cdots$ . We constructthe generating functions $f_{m}$ where the co-efficient of $z^{n}$ equals to $t’(m, n)$ whenever$m\geq n$ .

Case $m=1$ and no monominoes: $B=$$1+z^{2}B$ (empty or starts with a horizontaltile). This yields $B=1/(1-z^{2})$ .

Case $m=1$ and one monomino: $A=$$zB+z^{2}A$ (starts with a monomino or a hor-izontal tile). Solving for $A$ yields

$f_{1}= \frac{z}{(1-z^{2})^{2}}$ .

In what follows we only consider $m\geq 3$ ,$m$ odd.

Computing $c:A=1+z^{m-1}A+z^{m+1}A$(from formula in the previous section). Thisyields $A=1/(1-z^{m-1}-z^{m+1})$ .

Computing $t_{2}:B=4z^{m-2}A+4z^{m}A$ (fromformula in the previous section). This yields$B=4(z^{m-2}+z^{m})/(1-z^{m-1}-z^{m+1})$ for$m\geq 5$ . omitting $4z^{m-2}$ for $m=3$ , we get$B=4z^{m}/(1-z^{m-1}-z^{m+1})$ .

Computing $t_{3}:C=z^{m-1}C+z^{m+1}C+$$2zA+2z^{m}A$ (from formula in the previoussection). This yields $C=2(z+z^{m})/(1-$$z^{m-1}-z^{m+1})^{2}$ .

Finally, $B+C$ yields: for $m\geq 5,$ $m$ odd

$f_{m}= \frac{1}{(1-z^{m-1}-z^{m+1})^{2}}\cdot(2z+4z^{m-2}+$

$6z^{m}-4z^{2m-3}-8z^{2m-1}-4z^{2m+1})$ ,

while for $m=3$ the generating function is

$f_{3}= \frac{2z+6z^{m}-4z^{2m-1}-4z^{2m+1}}{(1-z^{m-1}-z^{m+1})^{2}}$ .

We recall that for $m=1$ we had

$f_{1}= \frac{z}{(1-z^{2})^{2}}$ .

8 Multiple monominoesWe have given the characterizations oftiling rectangles by dominoes and a singlemonomino. Note that it would require atleast two monominoes to tile a rectangle 7by 10. It would also require at least threemonominoes to tile a rectangle 9 by 13.However, the variety of tilings with arbi-trary number of monominoes is quite “wild”in sense that such tilings cannot be easilydecomposed, see Figure 11; therefore, mostresults presented here do not generalize toarbitrary number of monominoes, the tech-niques used here are not applicable, and it isexpected that any characterization or enu-meration of them would be much more com-plicated.

$\prod_{\square }\prod_{\square \prod_{\square }^{\fbox{}}}$

$\prod_{-,\coprod\coprod_{\coprod_{\Pi}}^{\square ^{\Pi}}\coprod_{\square }\coprod^{\square }\coprod\coprod_{\square \square }^{\square ^{\Pi}}}\square \square \square \square _{\square }\square \square \coprod_{\square _{\square }}\square \coprod_{\square \square ^{\square }}\square _{\square }\coprod\coprod_{\coprod_{\square }}\square \coprod_{\square \square }\coprod^{\coprod}\coprod\coprod\coprod\square _{\square _{\square }^{\square }\square _{\square }^{\square }\square }^{\Pi\prod_{\square }\prod_{\square }}\coprod_{\square ^{\square }}\square \square \square \square \square ^{\square }\square \square$

Figure 11: Tiling 7 by 23 with 5 monomi-noes

References[1] D. Hickerson: Filling rectangular

rooms with tatami mats, 2002.http: $//www$ . research. att. $com/$

$\sim njas/sequences/a068920$ . txt

[2] F. Ruskey, J. Woodcock: Count-ing Fixed-Height Tatami Tilings.The Electronic Joumal of Combi-natorics 16(1), 2009, R126, 20 pp.,http: $//www$ . combinatorics. $org/$

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