%I A093695
%S A093695 0,0,2,4,10,18,34,56,94,146,228,340,506,730,1050,1476,2066,2844,3896,
%T A093695 5268,7090,9442,12518,16454,21534,27980,36210,46572,59674,76056,96594,
%U A093695 122106,153852,193048,241492,300974,374038,463286,572304,704826,865874
%N A093695 Number of one-element transitions among partitions of the integer n for 
               unlabeled parts.
%C A093695 a(n) = Sum_p=1^P(n) Sum_i=1^D(p) Sum_j=1^D(p) 1 [subject to: i <> j and 
               d(i,p) <= d(j,p) and d(i,p) <> d(i-1,p) (if i > 1) and d(j,i) <> 
               d(j-1,i) (if j > 1 and if d(j-1,p) has given a contribution to the 
               sum) ]; 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.
%C A093695 See the corresponding formula for a(n) for the labeled case A094533.
%H A093695 T. Wieder, <a href="http://www.thomas-wieder.privat.t-online.de">Home 
               Page</a>.
%H A093695 T. Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder">(old) Home 
               Page</a>.
%F A093695 a(n) = Sum_i=1^P(n+1) S(i, n+1)^2 - S(i, n+1), where P(n+1) is the number 
               of integer partitions of n+1 and S(i, n+1) is the number of digits 
               in the set of digits of the i-th partition of n+1. (E.g. the partition 
               [1111233] has the set of digits {1, 2, 3} and would contribute 3^2 
               - 3 = 6 to the sum.)
%e A093695 In the unlabeled case we have 10 one-element transitions among all partitions 
               of n=4: [1,1,1,1] -> [1,1,2]; [1,1,2] -> [2,2]; [1,1,2] -> [1,3]; 
               [2,2] -> [1,3]; [1,3] -> [4] and [1,1,2] -> [1,1,1,1]; [2,2] -> [1,
               1,2]; [1,3] -> [1,1,2]; [1,3] -> [2,2]; [4] -> [1,3].
%e A093695 n=5:
%e A093695 partition number p=1 is [1,1,1,1,1], digits d(1,1)=1, d(2,1)=1 contribute 
               1;
%e A093695 partition number p=2 is [1,1,1,2], digits d(1,1)=1, d(2,2)=1 contribute 
               1, digits d(1,2)=2, d(4,2)=2 contribute 1;
%e A093695 partition number p=3 is [1,2,2], digits d(1,3)=1, d(2,3)=2 contribute 
               1, digits d(2,3)=2, d(3,3)=2 contribute 1;
%e A093695 partition number p=4 is [1,1,3], digits d(1,4)=1, d(2,4)=1 contribute 
               1, digits d(1,4)=1, d(3,4)=3 contribute 1;
%e A093695 partition number p=5 is [2,3], digits d(1,5)=2, d(2,5)=3 contribute 1;
%e A093695 partition number p=6 is [1,4], digits d(1,6)=1, d(2,6)=4 contribute 1;
%e A093695 partition number p=7 is [5], digits d(1,7)=5 contributes 0;
%e A093695 ==> a(5)=2*9=18 (factor 2 if we accept up and down transitions).
%e A093695 a(5) = 18 because the 11 partitions of n=5+1=6 have the following sets 
               of digits:
%e A093695 {1} contributes 0, {1, 2} contributes 2, {1, 2} contributes 2,
%e A093695 {2} contributes 0, {1, 3} contributes 2, {1, 2, 3} contributes 6,
%e A093695 {3} contributes 0, {1, 4} contributes 2, {2, 4} contributes 2,
%e A093695 {1, 5} contributes 2, {6} contributes 0,
%e A093695 which gives 0 + 2 + 2 + 0 + 2 + 6 + 0 + 2 + 2 + 2 + 0 = 18.
%p A093695 A093695 := proc(n::integer) local a,ndxp,ListOfPartitions,APartition,
               PartOfAPartition,SetOfParts, iverbose; with(combinat): iverbose:=1; 
               ListOfPartitions:=partition(n+1); a:=0; for ndxp from 1 to nops(ListOfPartitions) 
               do APartition := ListOfPartitions[ndxp]; SetOfParts := convert(APartition,
               set); a := a + nops(SetOfParts)^2 - nops(SetOfParts); if iverbose 
               = 1 then print ("ndxp, SetOfParts, nops(SetOfParts)^2 - nops(SetOfParts): 
               ", ndxp,SetOfParts,nops(SetOfParts)^2 - nops(SetOfParts)); fi; # 
               End of do-loop *** ndxp ***. end do; print("n, a(n):",n,a); end proc;
%t A093695 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) a[n_] 
               := Block[{p = Partitions[n + 1], l = PartitionsP[n + 1]}, Sum[ Length[ 
               Union[ p[[k]]]]^2 - Length[ Union[ p[[k]] ]], {k, l}]]; Table[ a[n], 
               {n, 2, 40}] (from Robert G. Wilson v Jul 13 2004)
%Y A093695 Cf. A093694, A089378.
%Y A093695 Cf. A094533.
%Y A093695 Sequence in context: A057491 A005541 A045955 this_sequence A104723 A079162 
               A043330
%Y A093695 Adjacent sequences: A093692 A093693 A093694 this_sequence A093696 A093697 
               A093698
%K A093695 nonn
%O A093695 0,3
%A A093695 Thomas Wieder (wieder.thomas(AT)t-online.de), Apr 10 2004
%E A093695 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 13 2004