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A094533
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Number of one-element transitions among partitions of the integer n for labeled parts.
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3
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0, 0, 2, 8, 22, 48, 98, 178, 316, 524, 856, 1334, 2066, 3084, 4578, 6626, 9530, 13434, 18854, 26022, 35764, 48520, 65526, 87550, 116536, 153674, 201906, 263258, 342006, 441366, 567754, 726032, 925588, 1174010, 1484664, 1869072, 2346586, 2934044
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OFFSET
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0,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..700
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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;
# second Maple program
b:= proc(n, i) option remember; local f, g;
if n=0 then [1, [1]] elif i<1 then [0, [0]]
else f:= b(n, i-1); g:= `if`(i>n, [0, [0]], b(n-i, i));
[f[1]+g[1], zip((x, y)-> x+y, f[2], [0, g[2][]], 0)]
fi
end:
a:= proc(n) local l; l:= b(n, n)[2];
add (l[t+1]*t*(t-1), t=1..nops(l)-1)
end:
seq (a(n), n=0..50); # Alois P. Heinz, Apr 05 2012
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MATHEMATICA
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(* 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)
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CROSSREFS
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Cf. A093695.
Sequence in context: A137101 A212970 A212683 * A006696 A094939 A006732
Adjacent sequences: A094530 A094531 A094532 * A094534 A094535 A094536
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KEYWORD
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nonn
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AUTHOR
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Thomas Wieder, Jun 05 2004
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EXTENSIONS
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More terms from Robert G. Wilson v, Jul 13 2004
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STATUS
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approved
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