A257493 -id:A257493 - cp's OEIS frontend

A003439 Number of 6 X 6 stochastic matrices of integers: all rows and columns sum to n.

1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, 8575979362560, 94459713879600, 842286559093240, 6292583664553881, 40447642842118656, 228438173705550566, 1152877640765297760, 5271278793334883190, 22085628572718605376, 85604721304213863531

Offset: 0

N. J. A. Sloane

nonn

Matthias Beck and Dennis Pixton, The Ehrhart Polynomial of the Birkhoff Polytope, Discrete & Computational Geometry, 30(4)(2003), 623-637.
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Row n=6 of A257493.

Cf. A005467.

a(n) = Sum_{j=0..10} A005467(j) * binomial(5+j+n, 5+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Melissa Erdmann (merdmann(AT)nebrwesleyan.edu), May 07 2009
Offset changed to 0 by Alois P. Heinz, Apr 26 2015
Name clarified by Charles R Greathouse IV, Mar 03 2018

A172806 Number of n X n of nonnegative integers with all row and column sums equal to 4.

Original entry on oeis.org

1, 1, 5, 120, 10147, 2224955, 1047649905, 936670590450, 1455918295922650, 3680232136895819610, 14356628851597700179050, 82857993930808028192521800, 683327637694741065563262206250, 7821620120684573354895941635688250

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n = 0..56
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
M. L. Stein and P. R. Stein, Enumeration of Linear Graphs and Connected Linear Graphs up to p = 18 Points. Report LA-3775, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Oct 1967.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Cf. A000681, A001500, A172862, A172894, A172919, A172944, A172958.

Column k=4 of A257493.

A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 12, 1, 7, 22, 9, 1, 64, 1, 34, 121, 11, 1, 525, 2, 13, 407, 2022, 1, 801, 1, 10163, 1036, 17, 6211, 41735, 1, 19, 2212, 285784, 1, 3822, 1, 381446, 2229142, 23, 1, 1189540, 2, 22069276, 7261, 2309410, 1, 20943183, 164176641

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

Also the number of non-normal semi-magic squares with positive integer entries summing to n.

			The a(12) = 12 matrices:
  [12]
.
  [1 5] [5 1] [2 4] [4 2] [3 3]
  [5 1] [1 5] [4 2] [2 4] [3 3]
.
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200 (terms 0..59 from Chai Wah Wu)

Cf. A000290, A006052, A007016, A103198, A120732, A257493, A321719, A321722, A323349, A323523, A323524, A323529.

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]]}]];
ptnsqrs[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),And[SameQ@@Length/@#,Length[#]==0||Length[#]==Length[First[#]]]&];
Table[Sum[Length[Select[ptnsqrs[Times@@Prime/@y],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{y,IntegerPartitions[n]}],{n,10}]

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(n) = Sum_{d|n, d<=n/d} A257493(d, n/d-d) for n > 0. - Andrew Howroyd, Apr 10 2020

a(16)-a(55) from Chai Wah Wu, Jan 20 2019

A008552 Number of 7 X 7 stochastic matrices of integers.

Original entry on oeis.org

1, 5040, 9135630, 4662857360, 936670590450, 94161778046406, 5562418293759978, 215717608046511873, 5945968652327831925, 123538613356253145400, 2023270039486328373811, 27046306550096288483238, 303378141987182515342992, 2920054336492521720572276

Offset: 0

John Mount (jmount(AT)msri.org)

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row n=7 of A257493.

Offset changed to 0 by Alois P. Heinz, Apr 26 2015

A160318 Number of 8 X 8 stochastic matrices of integers.

Original entry on oeis.org

1, 40320, 545007960, 1579060246400, 1455918295922650, 569304690994400256, 114601242382721619224, 13590707419428422843904, 1046591482728407939338275, 56272722406349235035916800, 2233160342369825596702148720, 68316292103293669997188919040

Offset: 0

Melissa Erdmann (merdmann(AT)nebrwesleyan.edu), May 08 2009

nonn

Matthias Beck and Dennis Pixton, The Ehrhart Polynomial of the Birkhoff Polytope, Discrete & Computational Geometry, 30(4)(2003), 623-637.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9

Row n=8 of A257493.

A160319 Number of 9 X 9 stochastic matrices of integers.

Original entry on oeis.org

1, 362880, 41514583320, 772200774683520, 3680232136895819610, 6274236760589024662176, 4801114002903931560293544, 1933216160887575268614599040, 459761347800901006933211075259, 70417932475495769964322670258947, 7424168163091445250817494013145952

Offset: 0

Melissa Erdmann (merdmann(AT)nebrwesleyan.edu), May 08 2009

nonn

Matthias Beck and Dennis Pixton, The Ehrhart Polynomial of the Birkhoff Polytope, Discrete & Computational Geometry, 30(4)(2003), 623-637.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9

Row n=9 of A257493.

A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 14, 87, 148, 87, 14, 1, 1, 103, 4306, 63110, 388615, 1115068, 1575669, 1115068, 388615, 63110, 4306, 103, 1, 1, 694, 184015, 15902580, 567296265, 9816969306, 91422589980, 490333468494, 1583419977390, 3166404385990, 3982599815746, 3166404385990

Offset: 1

N. J. A. Sloane, Jul 03 2015

The n-th row of A257493 is a polynomial of degree (n-1)^2. This triangle gives the coefficients of the numerator of the generating functions for A257493 with denominators being (1-x)^(1+(n-1)^2). - Andrew Howroyd, Apr 11 2020

			Triangle begins:
  1;
  1;
  1,1,1;
  1,14,87,148,87,14,1;
  1,103,4306,63110,388615,1115068,1575669,1115068,388615,63110,4306,103,1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..177 (rows 1..9)
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477, doi:10.1137/0204040.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)

Row sums are A037302.

Cf. A005466, A005467, A257493.

T(n,k) = Sum_{i=0..k} A257493(n, k-i)*(-1)^i*binomial(1+(n-1)^2,i). - Andrew Howroyd, Apr 11 2020

a(1)=1 prepended and terms a(26) and beyond from Andrew Howroyd, Apr 11 2020

cp's OEIS Frontend

A003439 Number of 6 X 6 stochastic matrices of integers: all rows and columns sum to n.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A172806 Number of n X n of nonnegative integers with all row and column sums equal to 4.

Views

Author

Keywords

Links

Crossrefs

A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A008552 Number of 7 X 7 stochastic matrices of integers.

Views

Author

Keywords

Links

Crossrefs

Extensions

A160318 Number of 8 X 8 stochastic matrices of integers.

Views

Author

Keywords

References

Links

Crossrefs

A160319 Number of 9 X 9 stochastic matrices of integers.

Views

Author

Keywords

References

Links

Crossrefs

A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions