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A211992
Triangle read by rows in which row n lists the partitions of n in colexicographic order.
77
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
OFFSET
1,4
COMMENTS
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].
Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - Omar E. Pol, Aug 24 2013
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
EXAMPLE
From Omar E. Pol, Aug 24 2013: (Start)
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)
From Gus Wiseman, May 10 2020: (Start)
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)
MATHEMATICA
colex[f_, c_]:=OrderedQ[PadRight[{Reverse[f], Reverse[c]}]];
Join@@Table[Sort[IntegerPartitions[n], colex], {n, 0, 6}] (* Gus Wiseman, May 10 2020 *)
PROG
(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 A026791 print partition: */
\\ 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) );
\\ Joerg Arndt, Sep 02 2013
CROSSREFS
The graded reversed version is A026792.
The length-sensitive refinement is A036037.
The version for reversed partitions is A080576.
Partition lengths are A193173.
Partition maxima are A194546.
Partition minima are A196931.
The version for compositions is A228525.
The Heinz numbers of these partitions are A334437.
Sequence in context: A093993 A193073 A228100 * A182937 A340035 A386016
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 18 2012
STATUS
approved