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A211992
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Triangle read by rows in which row n lists the partitions of n in colexicographic order.
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77
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1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 3, 2, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 2, 1, 5, 1, 2, 2, 2, 4, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 5, 1, 1, 2, 2, 2, 1, 4, 2, 1, 3, 3, 1, 6, 1, 3, 2, 2, 5, 2, 4, 3, 7
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OFFSET
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1,4
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COMMENTS
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The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].
The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - Joerg Arndt, Sep 02 2013
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LINKS
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EXAMPLE
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Illustration of initial terms:
-----------------------------------------
n Diagram Partition
-----------------------------------------
. _
1 |_| 1;
. _ _
2 |_| | 1, 1,
2 |_ _| 2;
. _ _ _
3 |_| | | 1, 1, 1,
3 |_ _| | 2, 1,
3 |_ _ _| 3;
. _ _ _ _
4 |_| | | | 1, 1, 1, 1,
4 |_ _| | | 2, 1, 1,
4 |_ _ _| | 3, 1,
4 |_ _| | 2, 2,
4 |_ _ _ _| 4;
. _ _ _ _ _
5 |_| | | | | 1, 1, 1, 1, 1,
5 |_ _| | | | 2, 1, 1, 1,
5 |_ _ _| | | 3, 1, 1,
5 |_ _| | | 2, 2, 1,
5 |_ _ _ _| | 4, 1,
5 |_ _ _| | 3, 2,
5 |_ _ _ _ _| 5;
. _ _ _ _ _ _
6 |_| | | | | | 1, 1, 1, 1, 1, 1,
6 |_ _| | | | | 2, 1, 1, 1, 1,
6 |_ _ _| | | | 3, 1, 1, 1,
6 |_ _| | | | 2, 2, 1, 1,
6 |_ _ _ _| | | 4, 1, 1,
6 |_ _ _| | | 3, 2, 1,
6 |_ _ _ _ _| | 5, 1,
6 |_ _| | | 2, 2, 2,
6 |_ _ _ _| | 4, 2,
6 |_ _ _| | 3, 3,
6 |_ _ _ _ _ _| 6;
...
Triangle begins:
[1];
[1,1], [2];
[1,1,1], [2,1], [3];
[1,1,1,1], [2,1,1], [3,1], [2,2], [4];
[1,1,1,1,1], [2,1,1,1], [3,1,1], [2,2,1], [4,1], [3,2], [5];
[1,1,1,1,1,1], [2,1,1,1,1], [3,1,1,1], [2,2,1,1], [4,1,1], [3,2,1], [5,1], [2,2,2], [4,2], [3,3], [6];
(End)
The triangle with partitions shown as Heinz numbers (A334437) begins:
1
2
4 3
8 6 5
16 12 10 9 7
32 24 20 18 14 15 11
64 48 40 36 28 30 22 27 21 25 13
128 96 80 72 56 60 44 54 42 50 26 45 33 35 17
(End)
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MATHEMATICA
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colex[f_, c_]:=OrderedQ[PadRight[{Reverse[f], Reverse[c]}]];
Join@@Table[Sort[IntegerPartitions[n], colex], {n, 0, 6}] (* Gus Wiseman, May 10 2020 *)
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PROG
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(PARI)
gen_part(n)=
{ /* Generate partitions of n as weakly increasing lists (order is lex): */
my(ct = 0);
my(m, pt);
my(x, y);
\\ init:
my( a = vector( n + (n<=1) ) );
a[1] = 0; a[2] = n; m = 2;
while ( m!=1,
y = a[m] - 1;
m -= 1;
x = a[m] + 1;
while ( x<=y,
a[m] = x;
y = y - x;
m += 1;
);
a[m] = x + y;
pt = vector(m, j, a[j]);
\\ for (j=1, m, print1(pt[j], ", ") );
/* for A211992 print partition as weakly decreasing list (order is colex): */
forstep (j=m, 1, -1, print1(pt[j], ", ") );
ct += 1;
);
return(ct);
}
for(n=1, 10, gen_part(n) );
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CROSSREFS
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The graded reversed version is A026792.
The length-sensitive refinement is A036037.
The version for reversed partitions is A080576.
The version for compositions is A228525.
The Heinz numbers of these partitions are A334437.
Cf. A036036, A080577, A193073, A228100, A296150, A331581, A334301, A334302, A334436, A334439, A334442.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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