A005467
6 X 6 stochastic matrices of integers.
Original entry on oeis.org
1, 714, 196677, 18941310, 809451144, 17914693608, 223688514048, 1633645276848, 6907466271384, 15642484909560, 14666561365176
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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.
- I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
- I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
- 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]
A052280
Number of 4 X 4 stochastic matrices under row and column permutations.
Original entry on oeis.org
1, 1, 5, 12, 43, 106, 321, 787, 1960, 4354, 9386, 18790, 36362, 66789, 118936, 203840, 340195, 551192, 873343, 1351457, 2052221, 3056798, 4480565, 6462678, 9194098, 12902867, 17892986, 24524478, 33265476, 44666016, 59426834, 78364873, 102502765, 133024660, 171390035, 219278224
Offset: 0
There are 5 nonisomorphic 4 X 4 matrices with row and column sums 2:
[0 0 0 2] [0 0 0 2] [0 0 0 2] [0 0 1 1] [0 0 1 1]
[0 0 2 0] [0 0 2 0] [0 1 1 0] [0 0 1 1] [0 1 0 1]
[0 2 0 0] [1 1 0 0] [1 0 1 0] [1 1 0 0] [1 0 1 0]
[2 0 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0] [1 1 0 0]
A075754
Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.
Original entry on oeis.org
1, 720, 3110940, 24046189440, 315031400802720, 6736218287430460752, 226885231700215713535680, 11649337108041078980732943360, 885282776210120715086715619724160, 96986285294151066094112970262797953280
Offset: 5
Michel Buffet (buffet(AT)engref.fr), Oct 08 2002
- B. D. McKay, Applications of a technique for labeled enumeration, Congressus Numerantium, 40 (1983) 207-221.
- Vaclav Kotesovec, Table of n, a(n) for n = 5..61, (computed with program by Doron Zeilberger, see link below)
- B. D. McKay, 0-1 matrices with constant row and column sums
- E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums.
- Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses, Maple package Bipartite.
- 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]
- Index entries for sequences related to binary matrices
A172862
Number of n X n 0..5 arrays with row sums 5 and column sums 5.
Original entry on oeis.org
1, 1, 6, 231, 40176, 22069251, 30767936616, 94161778046406, 569304690994400256, 6274236760589024662176, 118285830126660123474844752, 3623440212198461411381072575512, 172850452498398420310370097345242112
Offset: 0
A003439
Number of 6 X 6 stochastic matrices of integers: all rows and columns sum to n.
Original entry on oeis.org
1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, 8575979362560, 94459713879600, 842286559093240, 6292583664553881, 40447642842118656, 228438173705550566, 1152877640765297760, 5271278793334883190, 22085628572718605376, 85604721304213863531
Offset: 0
- 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]
More terms from Melissa Erdmann (merdmann(AT)nebrwesleyan.edu), May 07 2009
A005466
5 X 5 stochastic matrices of integers.
Original entry on oeis.org
1, 115, 5390, 101275, 858650, 3309025, 4718075
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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.
- I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
- I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
- 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]
A052281
Number of 4 X 4 symmetric stochastic matrices under row and column permutations.
Original entry on oeis.org
1, 1, 3, 6, 16, 29, 62, 107, 195, 320, 522, 804, 1234, 1804, 2626, 3700, 5155, 7038
Offset: 0
There are 6 nonisomorphic symmetric 4 X 4 matrices with row and column sums 3:
[0 0 0 3] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 0 1 2] [0 1 1 1]
[0 0 3 0] [0 0 2 1] [0 1 1 1] [0 1 2 0] [0 2 1 0] [1 0 1 1]
[0 3 0 0] [1 2 0 0] [1 1 1 0] [1 2 0 0] [1 1 0 1] [1 1 0 1]
[3 0 0 0] [2 1 0 0] [2 1 0 0] [2 0 0 1] [2 0 1 0] [1 1 1 0]
But, A333886 gives 6 other cases.
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, 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]
A123228
Sum of the n-th powers of the roots of the polynomial x^6 + 14x^5 + 87x^4 + 148x^3 + 87x^2 + 14x + 1.
Original entry on oeis.org
6, -14, 22, 466, -6714, 51346, -205418, -638414, 19787526, -195455054, 1126500502, -1636604654, -47878102074, 662684162386, -4965254864618, 19072814136946, 71067700116486, -1976406503675534, 19086772122105622, -107375947452919214, 128777308208472006, 4884916184617735186
Offset: 0
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Newt:=proc(f) local t1,t2,t3,t4; t1:=f; t2:=diff(f,x); t3:=expand(x^degree(t1,x)*subs(x=1/x,t1)); t4:=expand(x^degree(t2,x)*subs(x=1/x,t2)); factor(t4/t3); end;
t1:=1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6; Newt(t1); series(t1,x,50);
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polsym(x^6 + 14*x^5 + 87*x^4 + 148*x^3 + 87*x^2 + 14*x + 1, 30) \\ Charles R Greathouse IV, Jul 20 2016
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