A003316

[David W. Wilson]

%N A003316 Sum of lengths of longest increasing subsequences of all permutations
 of n elements.
%S A003316 1,3,12,58,335,2261,17465,152020,1473057,15730705,183571817,
%T A003316 2324298010,31737207026,464904410985,7272666016725,121007866402968

When I classify permutations by longest increasing subsequence, I get

     n     1     2     3     4     5     6     7 perms   sum
     1     1                                         1     1
     2     1     1                                   2     3
     3     1     4     1                             6    12
     4     1    16     6     1                      24    55
     5     1    69    41     8     1               120   299
     6     1   348   293    67    10     1         720  1900
     7     1  2016  2309   602    99    12     1  5040 13942

My table is in agreement with A008304, but my sums are not in agreement
with A003316 above.