# ENUMERATION OF POLYOMINOES INSCRIBED IN A RECTANGLE by ert634

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```									      ENUMERATION OF POLYOMINOES
INSCRIBED IN A RECTANGLE
Alain Goupil, Hugo Cloutier, Fathallah Nouboud
Université du Québec à Trois-Rivières

Polyomino inscribed   Polyomino with    Polyomino with
in a rectangle        minimal area      minimal +1 area

Problem. Enumerate polyominoes inscribed in a rectangle !

1
   Enumeration of polyominoes with minimal area
 area=b+k-1
 they are disposed along a diagonal of the rectangle

First decomposition.

minimal         =      hook       ×      stair          ×   hook

Building blocs :
hook(b,k)=1 (the corner cell is fixed)
xy
⇒ Hook(x,y)=1 +      \$     x b yk = 1 +
(1 " x )(1 " y)
b,k#1
#b + k " 2 &
stair(b,k) = %          (
\$ b -1 '
!               #b + k " 2 & b k       xy
⇒       Stair(x,y)= * %              ( x y =
\$ b -1 '           (1 " x " y)
b,k)1
!
Polyominoes on one diagonal :
#       xy        &2    xy
⇒       Pmin,\(x,y)= %1 +
!                             (
\$ (1 " x )(1 " y) ' (1 " x " y)

2
!
Polyominoes on two diagonals: Crosses

cross = hook × (hook – corner cell)
except for polyominoes on one row or one column

xy                   xy 2              x2y
⇒       Cross(x,y)=                             "                 "
2          2                 2
(1 " x ) (1 " y)           (1 " y)           (1 " x ) 2

Inclusion-exclusion.
!

Pmin(x,y)= Pmin,\(x,y)+Pmin,/(x,y)-cross(x,y)

⇒      Pmin(x,y)=      # pmin ( b,k )x b yk
b,k "1

#          xy         &2   2xy
#
xy             xy 2       x 2y (
&
%1 +                  (             "%                      "          "
\$    (1 " x )(1 " y ) ' (1 " x " y ) % (1 " x ) 2 (1 " y ) 2 (1 " y ) 2 (1 " x ) 2 (
!                          \$                                             '

!

3
Exact formulas :

#b + k " 2 &
Pmin(b,k)= 8 %          ( " bk " 2( b "1)( k "1) " 6
\$ b -1 '

if         n = number of cells
then Pmin(n) = number of polyominoes with n cells
!        inscribed in any rectangle of perimeter 2n+2
n
= # pmin ( b,n " b +1)
b =1
1# 3               &
= 2n+2 "      %n " n 2 +10n + 4 (
2\$                 '
!

n     z 2 (1 " 4z + 8z 2 " 6 z 3 + 4z 4 )
Pmin(z)=
!           # pmin ( n )z =
b,k "1                     (1 " z ) 4 (1 " 2z )

!                  !

4
Second decomposition :

minimal       =     hook       × corner polyomino

Corner polyominoes : inscribed polyominoes with min area
and one cell in a given corner of the rectangle.
#1                                if b = 1 or k = 1
pc(b,k)= \$
% pc ( b "1,k ) + pc ( b,k "1) +1 otherwise
#b + k " 2 &
= 2%          ( "1 for b,k ) 1
\$ b -1 '
!

!

5
   Polyominoes with min+1 area

Benches :

P is an inscribed polyomino of area min+1
⇔
P contains exactly one bench

To construct all min+1 polyominoes that contain a given
bench B:
1- Fix the position of the bench B in a b × k rectangle R.
2- Complete the bench into a polyomino with area min+1
in two opposite regions; f1--f2 or f3--f4.
3- Use inclusion-exclusion and remove polyominoes that
belong to both diagonals (i.e. hooks).

Pmin+1(B) =     f1 f2      +       f3 f4           -8t

To obtain all inscribed min+1 polyominoes :
4- sum over all benches B in the rectangle R.
6
Case 1. The bench B is in a corner.

P1(t,b,k)        = Corner polyomino + Hook,
#b + k " t " 2 &
= 2%              ( + 2( t "1)
\$ b-2          '

!

P2(t,b,k)           = Corner polyomino + Hook,
#b + k " t " 2 &
= 2%              ( + 2
\$ b-2          '
Proposition. The number g1(b,k) of polyominoes of area
min+1 inscribed in a bxk rectangle with a bench in any
corner of!the rectangle is
\$ k "1                ' \$ k "1                    '
g1( b,k ) = &4 # p1( t ,b,k ) + 4 ) + &4 # p2 ( t ,b,k ) + 2k )
&                     ) &                         )
% t=3                 ( % t=3                     (
\$ b "1                ' \$ b "1                    '
+ &4 # p1( t ,b,k ) + 4 ) + &4 # p2 ( t ,b,k ) + 2b )
&                     ) &                         )
% t=3                 ( % t=3                     (
*\$b + k - 4 ' \$b + k - 4 '-
= 16 ,&          )+&            )/ + 2k( 2k "1) + 2b( 2b "1) "72
+%  b - 1 ( % k - 1 (.
7
Case 2. The bench is on one side of the rectangle and not
in a corner.

Proposition. The number g2(b,k) of polyominoes of area
min+1 inscribed in a b×k rectangle with a bench touching
exactly one side of the rectangle is

("b + k - 4 % "b + k - 4 %+
g2 ( b,k ) = 32*\$            '+\$          '- +
)# b        & # k        &,
( "b + k - 4 % "b + k - 4 % "b + k - 4 %+
8*10 \$           '+\$          '+\$         '- +
)    # b - 2 & # b - 1 & # k - 1 &,
4 3                                     164
( b + k 3 ) . 28( b2 + k 2 ) . 48bk +      (b + k)
3                                        3
+ 4( bk 2 + b2k ) +144

!

8
Case 3. The bench touches no side of the rectangle.

Proposition. The number g3(b,k) of polyominoes of area
min+1 inscribed in a b×k rectangle with a bench
touching no side of the rectangle is
8       )#b + k - 4 & #b + k - 4 &,
g3 ( b,k ) =
3
[ "12+%
*\$ b
(+%
' \$ k
(. +
'-
)#b + k - 4 & #b + k - 4 &,      #b + k - 4 &
6( b + k " 6 )+%           ( +%        (. " 60 %          (
*\$ b - 1 ' \$ k - 1 '-            \$ b- 2 '
#b + k - 2 &       3    3        2      2
+18%             ( " ( b + k ) +15( b + k )
\$ b-1 '
,
" 6( bk 2 + b2k ) " 48bk " 56( b + k ) + 24.
-

Case 4. 2×2 benches.
!
Proposition. The number p2"2 ( b,k ) of polyominoes of
area min+1 inscribed in a bxk rectangle with a bench
touching no side of the rectangle is
0                          !
2
24(b + k - 4)                           if b = 2,k " 3 or k = 2,b " 3
2 *#                                                 -
2 b + k - 4&       #b + k - 4 &     #b + k - 4 &
18,%          ( + 2%          ( + 2%           ( ) 3 / if b = 3 or k = 3
2 +\$ b - 2 '       \$ b-1 '          \$ k-1 '          .
2 *##           & &                    -
28,%%b + k - 4 ( +1(( b + k ) 2 ) ) bk /               if b,k " 4
2 ,\$\$ b - 2 ' '
3 +                                    /
.

!                                                                                         9
All cases.

Theorem. For b,k≥3, the number pmin+1(b,k) of
polyominoes of area min+1 inscribed in a b×k rectangle is
pmin+1(b,k)= g1(b,k)+ g2(b,k)+ g3(b,k)+ p2"2 ( b,k )

#b + k " 4 & 8( 2k 2 + 2kb + k "13k +13 ) #b + k " 4 &
= 8( b + k " 22 ) %          (+        !                    %          (
\$ b- 2 '                (k " 2)           \$ b-1 '
8( 2b2 + 2kb + b "13b +13 ) #b + k " 4 &     #b + k " 2 & 4 3    3
+                             %          ( + 48%          ( " (b +k )
(b " 2)           \$ k-1 '          \$ b-1 ' 3
266
"12( b2k + bk 2 ) +16( b2 + k 2 ) +72bk "       ( b + k ) +120
3
Corollary. For integers n≥4, the number pmin+1(n) of
polyominoes of area n inscribed in any rectangle of
!
perimeter 2n is given by
n"2
pmin+1( n ) =    # pmin+1( b,n " b)
b =2
22 ' 1 \$ 4
n \$4                                       '
=2 & +   n ) " &8n " 88n 3 + 430n 2 " 902n + 636 )
%5 5 ( 3 %                                    (

!

10
   Polyominoes with no loop (lattice trees)
and min+1 area.

Corollary. The number l min+1( b,k ) of lattice trees
inscribed in a b×k rectangle with area min+1 is
l min+1( b,k ) = f min+1( b,k ) " f 2#2 ( b,k )
!

Corollary. For integers n ≥ 5, the number l min+1( n ) of
!
lattice trees of area n inscribed in any rectangle of
perimeter 2n is given by
n"2                                  !
l min+1( n ) =    # l min+1( b,n " b)
b =2
2\$ 4                                '
= 2 n+1 n "1 "
(   )    &4n " 46n 3 + 227 n 2 " 473n + 318 )
3%                                  (

!

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Next …

 More recurrences, exact formulae, generating
functions

   Minimal 3D polyominoes

12

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