A322761 Irregular triangle read by rows in which n-th row lists all partitions of n, in graded reverse lexicographic ordering, using a compressed notation.

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

Offset

1,2

Comments

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.

Links

Alois P. Heinz, Rows n = 1..28, flattened

Example

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,
...
...

Maple

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

Mathematica

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 *)

Crossrefs

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 A365769

Adjacent sequences: A322758 A322759 A322760 * A322762 A322763 A322764

Keyword

nonn,look,tabf,base,fini

Author

N. J. A. Sloane, Dec 30 2018

Status

approved