OFFSET
1,2
COMMENTS
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 (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]
Written as a triangle; row n has length A006128(n); row sums give A066186. Also written as an irregular tetrahedron in which T(n,j,k) is the k-th largest part of the j-th partition of n; the sum of column k in the slice n is A181187(n,k); right border of the slices gives A182715. - Omar E. Pol, Mar 25 2012
The equivalent sequence for compositions (ordered partitions) is A228351. - Omar E. Pol, Sep 03 2013
This is the reverse-colexicographic order of integer partitions, or the reflected reverse-lexicographic order of reversed integer partitions. It is not reverse-lexicographic order (A080577), wherein we would have (3,1) before (2,2). - Gus Wiseman, May 12 2020
LINKS
Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order
EXAMPLE
E.g. the partitions of 3 (3,2+1,1+1+1) appear as the string 3,2,1,1,1,1.
So the list begins:
1
2, 1, 1,
3, 2, 1, 1, 1, 1,
4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1,
5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
...
From Omar E. Pol, Sep 03 2013: (Start)
Illustration of initial terms:
---------------------------------
n j Diagram Partition
---------------------------------
. _
1 1 |_| 1;
. _ _
2 1 |_ | 2,
2 2 |_|_| 1, 1;
. _ _ _
3 1 |_ _ | 3,
3 2 |_ | | 2, 1,
3 3 |_|_|_| 1, 1, 1;
. _ _ _ _
4 1 |_ _ | 4,
4 2 |_ _|_ | 2, 2,
4 3 |_ _ | | 3, 1,
4 4 |_ | | | 2, 1, 1,
4 5 |_|_|_|_| 1, 1, 1, 1;
...
(End)
From Gus Wiseman, May 12 2020: (Start)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows. Showing these partitions as their Heinz numbers gives A334436.
0
(1)
(2)(11)
(3)(21)(111)
(4)(22)(31)(211)(1111)
(5)(32)(41)(221)(311)(2111)(11111)
(6)(33)(42)(222)(51)(321)(411)(2211)(3111)(21111)(111111)
(End)
MATHEMATICA
revcolex[f_, c_]:=OrderedQ[PadRight[{Reverse[c], Reverse[f]}]];
Join@@Table[Sort[IntegerPartitions[n], revcolex], {n, 0, 8}] (* reverse-colexicographic order, Gus Wiseman, May 10 2020 *)
- or -
revlex[f_, c_]:=OrderedQ[PadRight[{c, f}]];
Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n], revlex], {n, 0, 8}] (* reflected reverse-lexicographic order, Gus Wiseman, May 12 2020 *)
PROG
(PARI) A026794_rows(n) = my(v1 = vector(n, i, vector(i\2, j, 0))); if(n>1, v1[2][1] = 1); for(i=3, n, v1[i][1] = vecsum(v1[i-1])+1; for(j=2, i\2, v1[i][j] = v1[i-1][j-1] - if((j-1)>((i-j)\2), 0, if((i-j)==(j-1), 1, v1[i-j][j-1])))); v1
partition(n, k) = my(v1 = A026794_rows(n), A = n, B = k-1, C, D, E, v2 = []); if(k<=0 || k>(vecsum(v1[n])+1), 0, while(B>0, C = 1; D = vecsum(v1[A]); E = D-v1[A][1]; while(!(B>E && B<=D), C++; D = E; E -= v1[A][C]); A -= C; B -= E+1; v2 = concat(v2, C)); v2 = concat(v2, A); v2 = Vecrev(v2)) \\ Mikhail Kurkov, Jan 28 2026
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
Terms 81st, 83rd and 84th corrected by Omar E. Pol, Aug 16 2009
STATUS
approved
