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A093694
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Number of one-element transitions from the partitions of n to the partitions of n+1 for labeled parts.
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10
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1, 2, 5, 9, 17, 27, 46, 69, 108, 158, 234, 331, 476, 657, 915, 1244, 1694, 2262, 3029, 3988, 5257, 6844, 8901, 11461, 14749, 18809, 23958, 30304, 38263, 48018, 60167, 74977, 93276, 115509, 142772, 175759, 215991, 264449, 323216, 393772, 478884
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OFFSET
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0,2
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COMMENTS
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For the unlabeled case, the number of one-element transitions from the partitions of n to the partitions of n+1 is given by A000070. Example: There are A000070(4) = 12 transitions from n=4 to n=5: [1111] -> [11111], [1111] -> [1112], [112] -> [1112], [112] -> 113], [112] -> [122], [13] -> [113], [13] -> [14], [13] -> [23], [22] -> [23], [22] -> [122], [4] -> [14], [4] -> [5].
a(n) is also the total number of parts in all partitions of the integer n+1 which contain at least one part 1.
Also partitions of n into 2 sorts of parts where all parts of the first sort precede all parts of the second sort; see example. [Joerg Arndt, Apr 28 2013]
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = Sum_k=1^p(n) (nops(p(k, n)) + 1), where p(n) is the number of partitions of n and nops(p(k, n)) is the number of parts in the k-th partition p(n, k) of n.
a(n) = Sum_k=1^p(n) nops(p(k, n)[subject to: at least one p(l, k, n) = 1]; p(n) = number of partitions of n, p(k, n) = k-th partition, p(l, k, n) = l-th part in the k-th partition p(k, n) of integer n.
G.f.: sum(n>=0, (n+1) * x^n / prod(k=1..n, 1-x^k ) ). - Joerg Arndt, Apr 17 2011
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EXAMPLE
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In the labeled case, we have 9 one-element transitions from all partitions of n=3 to the partitions of n+1=4: [1,1,1] -> [1,1,1,1]; [1,1,1] -> [1,1,2]; [1,1,1] -> [1,1,2]; [1,1,1] -> [1,1,2]; [1,2] -> [1,1,2]; [1,2] -> [1,3]; [1,2] -> [2,2]; [3] -> [1,3]; [3] -> [4].
For n = 3 we have the following partitions of 3+1 = 4 which contain at least one part 1: [1111], [112], [13] and these partitions contain 4 + 3 + 2 = 9 = a(3) parts.
There are a(4)=17 partitions of 4 into 2 sorts where all parts of the first sort precede all parts of the second sort. Here p:s stands for "part p of sort s":
01: [ 1:0 1:0 1:0 1:0 ]
02: [ 1:0 1:0 1:0 1:1 ]
03: [ 1:0 1:0 1:1 1:1 ]
04: [ 1:0 1:1 1:1 1:1 ]
05: [ 1:1 1:1 1:1 1:1 ]
06: [ 2:0 1:0 1:0 ]
07: [ 2:0 1:0 1:1 ]
08: [ 2:0 1:1 1:1 ]
09: [ 2:0 2:0 ]
10: [ 2:0 2:1 ]
11: [ 2:1 1:1 1:1 ]
12: [ 2:1 2:1 ]
13: [ 3:0 1:0 ]
14: [ 3:0 1:1 ]
15: [ 3:1 1:1 ]
16: [ 4:0 ]
17: [ 4:1 ]
- Joerg Arndt, Apr 28 2013
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MAPLE
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main := proc(n::integer) local a, ndxp, ListOfPartitions; with(combinat): with(ListTools): ListOfPartitions:=partition(n-1); a:=0; for ndxp from 1 to nops(ListOfPartitions) do if Occurrences(1, ListOfPartitions[ndxp]) > 0 then a:=a+nops(Flatten(ListOfPartitions[ndxp])); print("ndxp, Flatten(ListOfPartitions[ndxp]):", ndxp, Flatten(ListOfPartitions[ndxp])); print("ndxp, ListOfPartitions[ndxp], a:", ndxp, ListOfPartitions[ndxp], a); # End of if-clause *** Occurrences(1, ListOfPartitions[ndxp]) *** fi; end do; print("n, a(n):", n, a); end proc;
##
b:= proc(n, i) option remember; local x, y;
if n<=0 or i=0 then [0, 0]
elif i=1 then [1, n]
else x:= b(n, i-1);
y:= b(n-i, i);
[x[1]+y[1], x[2]+y[2]+y[1]]
fi
end:
a:= n-> b(n+1, n+1)[2]:
seq (a(n), n=0..100); # Alois P. Heinz, Apr 24, 2011
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MATHEMATICA
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Needs["DiscreteMath`Combinatorica`"]
f[n_] := Block[{l = Sort[ Flatten[ Partitions[n]]]}, Length[l] - Count[l, 1]]; g[n_] := (f[n] + Sum[PartitionsP[k], {k, 0, n}]); Table[ g[n], {n, 0, 40}]
(* Robert G. Wilson v, Jul 13 2004 *)
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CROSSREFS
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Cf. A000070, A093695, A089378.
Sequence in context: A062492 A165271 A139672 * A068006 A000097 A081996
Adjacent sequences: A093691 A093692 A093693 * A093695 A093696 A093697
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KEYWORD
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nonn
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AUTHOR
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Thomas Wieder, Apr 10 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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