A321654 -id:A321654 - cp's OEIS frontend

A068313 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

1, 4, 15, 82, 457, 3231, 24055, 209375, 1955288, 20455936, 229830841, 2828166755, 37228913365, 528635368980, 7990596990430, 128909374528433, 2202090635802581, 39837079499488151, 759320365206705013, 15234890522990662422, 320634889654149218205, 7068984425261215971205

Offset: 1

Axel Kohnert (axel.kohnert(AT)uni-bayreuth.de), Feb 25 2002

nonn

This is the sum over the matrix of base change from the elementary symmetric functions to the monomial symmetric functions.

			a(2) = 4 because there are 4 different 0-1 matrices of weight 2: 1 10 01 11,1, 01, 10.
From _Gus Wiseman_, Nov 15 2018: (Start)
The a(3) = 15 matrices:
  [1 1 1]
.
  [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
(End)

I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford 1979, p. 57.

Ludovic Schwob, Table of n, a(n) for n = 1..37
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 14.

Cf. A000219, A001970, A007716, A049311, A101370, A117433, A120733, A321646, A321652, A321653, A321654.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@T[prs2mat[#]]]]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)

Name changed by Gus Wiseman, Nov 15 2018

a(20) onwards from Ludovic Schwob, Oct 13 2023

A321718 Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.

Original entry on oeis.org

1, 1, 5, 9, 44, 123, 986, 5043, 45832, 366300, 3862429, 39916803, 495023832, 6227020803, 88549595295, 1308012377572, 21086922542349, 355687428096003, 6427700493998229, 121645100408832003, 2437658338007783347, 51091307195905020227, 1125098837523651728389, 25852016738884976640003, 620752163206546966698620, 15511210044577707492319496

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A coupled non-normal semi-magic rectangle is a nonnegative integer matrix with equal row sums and equal column sums. The common row sum may be different from the common column sum.

Rectangles must be of size k X m where k and m are divisors of n. This implies that a(p) = p! + 3 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. The 1 X 1 square is [p], the 1 X p and p X 1 rectangles are [1,...,1] and its transpose and the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

			The a(3) = 9 coupled semi-magic rectangles:
  [3] [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]

Max Alekseyev, Table of n, a(n) for n = 0..69
Wikipedia, Magic square

Cf. A006052, A120733, A271103, A319056, A321654.

Cf. A321717, A321719, A321720, A321721, A321722, A321724, A321724.

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[multsubs[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}]

a(p) = p! + 3 for p prime. a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

a(7)-a(15) from Chai Wah Wu, Jan 15 2019
a(16)-a(19) from Chai Wah Wu, Jan 16 2019
Terms a(20) onward from Max Alekseyev, Dec 04 2024

A321653 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 44, 72, 147, 381, 1405

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(5) = 14 matrices:
  [5] [4 1] [3 2]
.
  [4] [4 0] [3 1] [3 1] [3] [3 0] [3 0] [2 2] [2 1] [2 1] [1 2]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1] [2 0]

Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321652, A321654, A321655.

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[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#],Greater],OrderedQ[Total/@Transpose[prs2mat[#]],Greater]]&]],{n,6}]

A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 21, 25, 37, 53, 137, 153, 249, 337, 505, 845, 1085, 1497, 2061, 2785, 3661, 7589, 8849, 13329, 18033, 26017, 34225, 48773, 70805, 91977, 123765, 164761, 216373, 283205, 367913, 470889, 758793, 913825, 1264105, 1651613, 2251709, 2894793, 3927837

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(6) = 21 matrices:
  [6] [1 5] [5 1] [2 4] [4 2] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [2] [4]
  [5] [1] [4] [2]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.

b:= proc(n, i, t) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
       b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
     If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
     b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Jan 15 2019

A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020, 23378901, 257871081, 3132494380, 40693204728, 572089068459, 8566311524788, 137165829681775, 2327192535461323, 41865158805428687, 793982154675640340, 15863206077534914434, 332606431999260837036, 7309310804287502958322, 167896287022455809865568

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(3) = 19 matrices:
  [3] [2 1] [1 1 1]
.
  [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Ludovic Schwob, Table of n, a(n) for n = 0..39
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.

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[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]]]&]],{n,6}]

Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.

a(10) onwards from Ludovic Schwob, Aug 29 2023

cp's OEIS Frontend

A068313 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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A321718 Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.

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A321653 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.

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A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

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A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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