An odd problem:    Torsten Sillke, 18. Dec. 1997

Count the number of symmetric 0-1-matrices 
for a given row-sum-vector bound.

Given e = (e1,e2,...,en) = (1,1,...,1) and
some vector b = (b1,b2,...,bn) how many 
symmetric 0-1-matrices A of order n satisfy

  A * e <= b ?

If you look at the table of counts you will
notice that almost all values are even.
Is there an involution for each vector b
showing this? Looking at the table I see
that (e,b) is even in all cases.

I give that table with all b vector for which
the number of matrices is odd (order 6).

b vector    count

000000         1
000011         5
000112        19
000222        45
001111        43
001113        71
001223       205
002222       315
002233       481
003333       809
011114       275
011123       465
011222       699
011224       911
012234      2381
012333      2967
013334      4443
022224      3801
022233      4749
022244      5681
023344     11809
033334     15705
033444     20643
044444     28217
111111       499
111115      1115
111223      3771
111225      4135
111234      5367
112222      5763
112224      8467
112233     10501
112235     11811
112334     16243
113333     20825
113335     23963
122225     18809
122234     26073
122245     30155
122333     33373
122344     43265
123334     56733
123345     67781
123444     76365
133335     90521
133344    102705
133445    126679
144445    184693
222244     70081
222255     74969
222334     91541
222345    110969
223335    147413
223344    167649
223355    182749
233444    329121
233455    365715
234445    427599
244455    568077
333344    449395
333355    504099
333445    592077
333555    671383
344555   1106673
444444   1146419
444455   1342251
445555   1590753
555555   1914733

In addition we get a new sequence of numbers

 1, 2, 4, 10, 24, 68, 198, 656

%I A004251 M1250 
%S A004251 1,2,4,11,31,102,342,1213,4361,16016,59348,222117 
%N A004251 Graphical partitions with n points. 
%R A004251 CN 21 684 78. 
%O A004251 1,2 
%A A004251 njas 
%K A004251 nonn,more