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A322761
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Irregular triangle read by rows in which n-th row lists all partitions of n, in graded reverse lexicographic ordering, using a compressed notation.
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9
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1, 2, 11, 3, 21, 111, 4, 31, 22, 211, 1111, 5, 41, 32, 311, 221, 2111, 11111, 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111, 7, 61, 52, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 22111, 211111, 1111111
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Officially this is deprecated, since one cannot distinguish between (for example) parts which are 11 and parts which are 1,1. However, it is in common use and is included for completeness. See A036037, A080577, etc., for uncompressed versions.
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LINKS
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Alois P. Heinz, Rows n = 1..28, flattened
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EXAMPLE
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Triangle begins:
1,
2, 11,
3, 21, 111,
4, 31, 22, 211, 1111,
5, 41, 32, 311, 221, 2111, 11111,
6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111,
7, 61, 52, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 22111, 211111, 1111111,
...
...
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MAPLE
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b:= (n, i)-> `if`(n=0 or i=1, [cat(1$n)], [map(x->
cat(i, x), b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(parse, b(n$2))[]:
seq(T(n), n=1..10); # Alois P. Heinz, Dec 30 2018
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MATHEMATICA
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revlexsort[f_, c_] := OrderedQ[PadRight[{c, f}]];
Table[FromDigits /@ Sort[IntegerPartitions[n], revlexsort], {n, 1, 8}] // Flatten (* Jean-François Alcover, Oct 20 2020, after Gus Wiseman in A080577 *)
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CROSSREFS
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Cf. A000041 (number of terms in row n), A036037, A080577.
See also A006128.
First column gives A000027.
Last elements of rows give A000042.
Sequence in context: A127668 A261300 A068743 * A113736 A241596 A339616
Adjacent sequences: A322758 A322759 A322760 * A322762 A322763 A322764
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KEYWORD
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nonn,look,tabf,base,fini
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AUTHOR
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N. J. A. Sloane, Dec 30 2018
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STATUS
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approved
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