On the Number of Magic Squares Matthias Beck Moshe Cohen Jessica - PDF document

On the Number of “Magic Squares” Matthias Beck Moshe Cohen Jessica Cuomo Paul Gribelyuk SUNY Binghamton www.binghamton.edu/matthias

Semi–magic square: square matrix whose entries are nonnegative integers and whose row and column sums are equal Magic square: semi–magic square whose main diagonals add up to the row–column sum Symmetric magic square: magic square which is symmetric Pandiagonal magic square: semi–magic square whose pandiagonals add up to the row–column sum. H n ( t ) := # semi–magic squares M n ( t ) := # magic squares S n ( t ) := # symmetric magic squares P n ( t ) := # pandiagonal magic squares for n × n squares with row–column–(pan)diagonal sum t 2

Example: H 2 ( t ) = t + 1 � 1 if t is even, M 2 ( t ) = S 2 ( t ) = P 2 ( t ) = 0 if t is odd. Theorem (Macmahon, 1915) 8 t 4 + 3 4 t 3 + 15 8 t 2 + 9 H 3 ( t ) = 1 4 t + 1 � 2 9 t 2 + 2 3 t + 1 if 3 | t M 3 ( t ) = 0 otherwise Theorem (Stein–Stein, 1970) H n ( t ) is a polynomial in t of degree ( n − 1) 2 . Theorem (Ehrhart, Stanley, 1973) H n ( − n − t ) = ( − 1) deg( H n ) H n ( t ) H n ( − 1) = H n ( − 2) = · · · = H n ( − n + 1) = 0 3

Conditions for magic 3 × 3 squares: x 1 ≥ 0 x 2 ≥ 0 . . . x 9 ≥ 0 x 1 + x 2 + x 3 = t x 4 + x 5 + x 6 = t x 7 + x 8 + x 9 = t x 1 + x 4 + x 7 = t x 2 + x 5 + x 8 = t x 3 + x 6 + x 9 = t x 1 + x 5 + x 9 = t x 3 + x 5 + x 7 = t This system describes a polytope in R 9 . We are interested in counting integer points in this polytope. Geometrically, the “magic variable” t is a dilation parameter. 4

Dilate the d –dimensional rational polytope P by a positive integer t : t P := { tx : x ∈ P} and count the number of integer points (“lattice points”) in t P : � t P ∩ Z d � L P ( t ) := # Theorem (Ehrhart, 1960’s) L P ( t ) is a quasi- polynomial in t whose degree is the dimen- sion of P . The period of this quasipoly- nomial divides any common multiple of the denominators of the vertices of P . 5

A quasipolynomial is an expression c d ( t ) t d + · · · + c 1 ( t ) t + c 0 ( t ) , where c 0 , . . . , c d are periodic functions in t Example:  0 for t = 1 , 4 , 7 , 10 , . . .  0 for t = 2 , 5 , 8 , 11 , . . . M 3 ( t ) = 9 t 2 + 2 2 3 t + 1 for t = 3 , 6 , 9 , 12 , . . .  Theorem M n ( t ) , S n ( t ) , P n ( t ) are quasipoly- nomials in t with degrees deg( M n ) = n 2 − 2 n − 1 2 n 2 − 1 deg( S n ) = 1 2 n − 2 deg( P n ) = n 2 − 3 n + 2 6

Idea of proof: The conditions of the 3 × 3 magic square yield the linear system   x 1     1 0 0 1 0 0 1 0 0 t x 2   0 1 0 0 1 0 0 1 0 t       x 3       0 0 1 0 0 1 0 0 1 t       x 4       1 1 1 0 0 0 0 0 0 t       = x 5       0 0 0 1 1 1 0 0 0 t       x 6       0 0 0 0 0 0 1 1 1 t       x 7       1 0 0 0 1 0 0 0 1 t       x 8   0 0 1 0 1 0 1 0 0 t x 9 7

Denote by M ⋆ n ( t ) , S ⋆ n ( t ) , P ⋆ n ( t ) the counting functions for magic squares, symmetric, and pandiagonal magic squares, respectively, as before, but now with the restriction that the entries are positive integers. For a d –dimensional rational polytope P , define � t P int ∩ Z d � L ⋆ P ( t ) := # Theorem (Ehrhart–Macdonald reciprocity law, 1971) If P is a d –dimensional rational poly- tope homeomorphic to a d –manifold then L P ( − t ) = ( − 1) d L ⋆ P ( t ) . ⇓ 8

Proposition M n ( − t ) = ( − 1) n 2 − 2 n − 1 M ⋆ n ( t ) 1 2 n 2 − 1 2 n − 2 S ⋆ S n ( − t ) = ( − 1) n ( t ) P n ( − t ) = ( − 1) n 2 − 3 n +2 P ⋆ n ( t ) M ⋆ n ( t ) = M n ( t − n ) ⇓ etc . M ⋆ n (1) = · · · = M ⋆ n ( n − 1) = 0 Theorem M n ( − t ) = ( − 1) n 2 − 2 n − 1 M n ( t − n ) 1 2 n 2 − 1 2 n − 2 S n ( t − n ) S n ( − t ) = ( − 1) P n ( − t ) = ( − 1) n 2 − 3 n +2 P n ( t − n ) M n ( − 1) = · · · = M n ( − n + 1) = 0 S n ( − 1) = · · · = S n ( − n + 1) = 0 P n ( − 1) = · · · = P n ( − n + 1) = 0 9

Find vertices by setting some x k = 0 .   x 1     1 0 0 1 0 0 1 0 0 1 x 2     0 1 0 0 1 0 0 1 0 1 x 3             0 0 1 0 0 1 0 0 1 1 x 4             0 0 0 1 1 1 0 0 0 = 1 x 5             0 0 0 0 0 0 1 1 1 1 x 6             1 0 0 0 1 0 0 0 1 1 x 7         0 0 1 0 1 0 1 0 0 1 x 8   x 9 Vertices: � � � � 3 , 0 , 1 2 3 , 0 , 1 3 , 2 3 , 1 3 , 2 1 3 , 2 3 , 0 , 0 , 1 3 , 2 3 , 2 3 , 0 , 1 3 , 0 , , 3 � � � � 0 , 2 3 , 1 3 , 2 3 , 1 3 , 0 , 1 3 , 0 , 2 1 3 , 0 , 2 3 , 2 3 , 1 3 , 0 , 0 , 2 3 , 1 , 3 3 10

To interpolate a quasipolynomial of degree d and period p , we need to compute p ( d +1) values. Example: M 3 ( t ) has degree 2 and period 3 M 3 (1) = 0 M 3 (2) = 0 M 3 (3) = 5 M 3 (4) = 0 M 3 (5) = 0 M 3 (6) = 13 M 3 (7) = 0 M 3 (8) = 0 M 3 (9) = 25  0 for t = 1 , 4 , 7 , 10 , . . .  0 for t = 2 , 5 , 8 , 11 , . . . = ⇒ M 3 ( t ) = 9 t 2 + 2 2 3 t + 1 for t = 3 , 6 , 9 , 12 , . . .  11

� 2 9 t 2 + 2 3 t + 1 if 3 | t M 3 ( t ) = 0 otherwise � 2 3 t + 1 if 3 | t S 3 ( t ) = 0 otherwise 2 t 2 + 2 P 3 ( t ) = 1 3 + 1  480 t 7 + 7 240 t 6 + 89 480 t 5 + 11 1 16 t 4   30 t 3 + 38 15 t 2 + 71 + 49  30 t + 1 if t is even   M 4 ( t ) = 480 t 7 + 7 240 t 6 + 89 480 t 5 + 11 1 16 t 4   480 t 3 + 593 240 t 2 + 1051  + 779 480 t − 3 16 if t is odd   128 t 4 + 5 16 t 3 + t 2 + 3 5  2 t + 1 if t ≡ 0 (4)    128 t 4 + 5 16 t 3 + t 2 + 3 5 2 t + 7 S 4 ( t ) = 8 if t ≡ 2 (4)   0 if t is odd  1440 t 6 + 7 120 t 5 + 23 72 t 4 + t 3 7   180 t 2 + 31 + 341 P 4 ( t ) = 15 t + 1 if t is even 0 if t is odd  12

What about the vertices/periods? Conjecture For n ≥ 4 , M n , S n , and P n are not polynomials. Application/next step: integer solutions to transportation problem/polytope 13

H d n ( t ) := # semi–magic hypercubes of dimension d , size n , and axial sums t H d n ( t ) is a quasipolynomial in t . H 2 = ( n − 1) 2 � � Stein–Stein: deg n � � H d = ( n − 1) d Theorem deg n M d n ( t ) := # magic hypercubes of dimension d , size n , and axial/diagonal sums t M d n ( t ) is a quasipolynomial in t . Earlier Theorem: = n 2 − 2 n − 1 = ( n − 1) 2 − 2 M 2 � � deg n � � = ( n − 1) d − 2 d − 1 M d Conjecture deg n Conjecture For n, d ≥ 3 , H d n and M d n are not polynomials. 14