0,3

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a(n) = Sum_p=1^P(n) Sum_i=1^D(p) Sum_j=i^D(p) 1 [subject to: d(i, p) <= d(j, p) ]; P(n) = number of partitions of n, D(p) = number of digits in partition p, d(i, p) and d(j, p) = digits number i and j in partition p of integer n.

a(n) = Sum_i=1^P(n) p(i, n)^2 - p(i, n), where P(n) is the number of integer partitions of n and p(i, n) is the number of parts of the i-th partition of n.

In the labeled case we have 22 one-element transitions among all partitions of n=4:

[1,1,1,1] -> [1,1,2] arises 6 times (the first 1 added to the second 1 gives 2,

the first 1 added to the third 1 gives 2, the first 1 added to the fourth 1 gives 2, the second 1 added to the third 1 gives 2, the second 1 added to the fourth 1 gives 2, the third 1 added to the fourth 1 gives 2),

[1,1,2] -> [2,2] arises 1 times,

[1,1,2] -> [1,3] arises 2 times,

[2,2] -> [1,3] arises 1 times,

[1,3] -> [4] arises 1 time,

which gives 11 upwards transitions and 22 transitions in total if we include downwards transitions.

n=4: partition number p=1 is [1,1,1,1],

digits d(1,1)=1, d(2,1)=1 contribute 1,

digits d(1,1)=1, d(3,1)=1 contribute 1,

etc...

digits d(3,1)=1, d(4,1)=1 contribute 1,

(in total 6 contributions by [1,1,1,1]);

partition number p=2 is [1,1,2],

digits d(1,2)=1, d(2,2)=1 contribute 1,

digits d(1,2)=1, d(3,2)=2 contribute 1,

digits d(2,2)=1, d(3,2)=2 contribute 1;

partition number p=3 is [2,2],

digits d(1,3)=2, d(2,3)=2 contribute 1;

partition number p=4 is [1,3],

digits d(1,4)=1, d(2,4)=3 contribute 1;

partition number p=5 is [4],

digit d(1,5)=4 contributes 0;

main := proc(n::integer) local a, ndxp, ListOfPartitions, APartition, PartOfAPartition; with(combinat): ListOfPartitions:=partition(n); a:=0; for ndxp from 1 to nops(ListOfPartitions) do APartition := ListOfPartitions[ndxp]; a := a + nops(APartition)^2 - nops(APartition); end do; print("n, a(n):", n, a); end proc;

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) a[n_] := Block[{p = Partitions[n], l = PartitionsP[n]}, Sum[ Length[p[[k]]]^2 - Length[p[[k]]], {k, l}]]; Table[ a[n], {n, 2, 37}] (from Robert G. Wilson v Jul 13 2004)

Cf. A093695.

Sequence in context: A085077 A153827 A137101 this_sequence A006696 A094939 A006732

Adjacent sequences: A094530 A094531 A094532 this_sequence A094534 A094535 A094536

nonn

Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 05 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 13 2004