A332567 T(n,k) is the k-th partition of n in graded reverse lexicographic ordering (A080577) encoded as concatenation of parts which are represented in (zeroless) bijective base-9 numeration (A052382) and separated by zeros; triangle T(n,k), n >= 0, 1 <= k <= A000041(n), read by rows.

0, 1, 2, 101, 3, 201, 10101, 4, 301, 202, 20101, 1010101, 5, 401, 302, 30101, 20201, 2010101, 101010101, 6, 501, 402, 40101, 303, 30201, 3010101, 20202, 2020101, 201010101, 10101010101, 7, 601, 502, 50101, 403, 40201, 4010101, 30301, 30202, 3020101, 301010101

Offset

0,3

Comments

The encoding used here allows a lossless and human-readable compression of all partitions. To decode a term replace the zeros with commas and read the parts in bijective base 9.

The empty partition is encoded as 0.

Links

Alois P. Heinz, Rows n = 0..34, flattened

Wikipedia, Bijective numeration

Wikipedia, Partition (number theory)

Example

T(6,6) = 30201 encodes the 6th partition of 6: [3,2,1].
T(10,1) = 11 encodes the 1st partition of 10: [10].
T(23,23) = 18040101 encodes the 23rd partition of 23: [17,4,1,1].
Triangle T(n,k) begins:
   0;
   1;
   2, 101;
   3, 201, 10101;
   4, 301, 202, 20101, 1010101;
   5, 401, 302, 30101, 20201, 2010101, 101010101;
   6, 501, 402, 40101, 303, 30201, 3010101, 20202, 2020101, ...
   7, 601, 502, 50101, 403, 40201, 4010101, 30301, 30202, ...
   8, 701, 602, 60101, 503, 50201, 5010101, 404, 40301, 40202, ...
   9, 801, 702, 70101, 603, 60201, 6010101, 504, 50301, 50202, ...
  11, 901, 802, 80101, 703, 70201, 7010101, 604, 60301, 60202, ...
  ...

Maple

g:= proc(n) option remember; local d, m, l; m, l:= n, "";
      while m>0 do d:= irem(m, 9, 'm');
        if d=0 then d:=9; m:= m-1 fi; l:= d, l
      od; parse(cat(l))
    end:
b:= (n, i)-> `if`(n=0, [""], `if`(i<1, [], [map(x-> cat(
     0, g(i), x), b(n-i, min(n-i, i)))[], b(n, i-1)[]])):
T:= n-> map(x-> parse(cat(0, x)), b(n$2))[]:
seq(T(n), n=0..10);

Crossrefs

Column k=1 gives A052382 (for n>0).

Last row elements give A094028(n-1) (for n>0).

Cf. A000041, A063008, A080577, A129129, A322761.

Sequence in context: A121975 A102188 A225601 * A126135 A016034 A171988

Adjacent sequences: A332564 A332565 A332566 * A332568 A332569 A332570

Keyword

nonn,tabf,look,base

Author

Alois P. Heinz, Feb 16 2020

Status

approved