partitions of k 'Covering' partitions of (k-1)

[wouter meeussen]

define 'covers' as :
a partition x covers partition y iff
the Ferrers plot of x covers that of y,
or,
for x= x1,x2 .. x_u and y=y1,y2 .. y_v
u>=v and  x_i >= y_j for i=1..v

For all couples (x,y),
where x is a partition of k,
and y is a partition of k-1,
the count of couples where x 'covers' y seems to be A000070,

1,2,4,7,12,19,30,45,67,97,139,195,272,373 ...

Name:      Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
Comments:  Partitions of n into parts but there are two kinds of parts of size one.
           Also number of graphical forest partitions of 2n+2.
           Also number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1,
              a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto
              (n_nomoto(AT)yabumi.com), Jan 09 2002
           a(n) = count 2 for each partition of n, and 1 for each decrement. E.g. the
              partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2).
              2+3+2+3+2=12. This is related to the Ferrers representation. We can
              see that taking the Ferrers diagram for each partition of n, and adding a
              new * to all available columns, we generate each partition of n+1, but
              with repeats (A058884). - Jon Perry (perry(AT)globalnet.co.uk), Feb
              06 2004

Is your (Jon Perry's) statement on expanding Ferrers plots equivalent to my coverage counting?
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Analogously for the plane partitions:
Table[Count[
    Outer[coversplaneQ, planepartitions[k], planepartitions[k - 1], 1],
    True, -1], {k, 12}]
 =
{1, 3, 9, 21, 48, 102, 213, 421, 819, 1542, 2854, 5172}

but this does *not* equal the running sum of A000219 = A091360
1, 2, 5, 11, 24, 48, 96, 182, 342, 624, 1124, 1983
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Mathematica:
<< DiscreteMath`Combinatorica`

coversQ[parent_, child_] :=
  And [ Length[parent] >= Length[child] ,
    Min[Take[ parent, Length @ child] - child] >= 0]

Table[Count[Outer[coversQ, Partitions[k], Partitions[k - 1], 1],
    True, -1], {k, 16}]
--------------------------------------------------------------------------------
by the way,
the simple
Table[PartitionsP[n + 1]*PartitionsP[n], {n, 0, 36}]

{1, 2, 6, 15, 35, 77, 165, 330, 660, 1260, 2352, 4312, 7777, 13635, 23760,
40656, 68607, 114345, 188650, 307230, 496584, 793584, 1257510, 1976625,
3083850, 4769688, 7332360, 11191180, 16972670, 25582260, 38342568, 57123858,
84683907, 124860330, 183209730, 267551691, 388968349}

is not yet in EIS, and superseeker can't find it.
Should it be in?
--------------------------------------------------------------------------------

W.