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A321717 Number of non-normal (0,1) semi-magic rectangles with sum of all entries equal to n. 14
1, 1, 4, 8, 39, 122, 950, 5042, 45594, 366243, 3858148, 39916802, 494852628, 6227020802, 88543569808, 1308012219556, 21086562956045, 355687428096002, 6427672041650478, 121645100408832002 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A non-normal semi-magic rectangle is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Rectangles must be of size k X m where k and m are divisors of n and k*m >= n. This implies that a(p) = p! + 2 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. There are no 1 X 1 rectangle that satisfies the condition. The 1 X p and p X 1 rectangles are [1....1] and its transpose, the p X p rectangle are necessarily permutation matrices and there are p! permutation matrices of size p X p. It also shows that a(n) >= n! + 2 for n > 1. - Chai Wah Wu, Jan 13 2019

LINKS

Table of n, a(n) for n=0..19.

Wikipedia, Magic square

FORMULA

a(p) = p! + 2 for p prime. a(n) >= n! + 2 for n > 1. - Chai Wah Wu, Jan 13 2019

EXAMPLE

The a(3) = 8 semi-magic rectangles:

  [1 1 1]

.

  [1] [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]

  [1] [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]

  [1] [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

MATHEMATICA

prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];

multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];

Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]

CROSSREFS

Cf. A006052, A007016, A068313, A101370, A120733, A271103, A319056.

Cf. A321718, A321719, A321720, A321721, A321722, A321723, A321724, A321724.

Sequence in context: A032042 A032056 A214535 * A228755 A007215 A195677

Adjacent sequences:  A321714 A321715 A321716 * A321718 A321719 A321720

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7) from Chai Wah Wu, Jan 13 2019

a(8)-a(13) from Chai Wah Wu, Jan 14 2019

a(14)-a(15) from Chai Wah Wu, Jan 15 2019

a(16)-a(19) from Chai Wah Wu, Jan 16 2019

STATUS

approved

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Last modified August 22 07:54 EDT 2019. Contains 326172 sequences. (Running on oeis4.)