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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                                                 12

								
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