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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.

%I A332567 #22 Feb 28 2020 00:28:09
%S A332567 0,1,2,101,3,201,10101,4,301,202,20101,1010101,5,401,302,30101,20201,
%T A332567 2010101,101010101,6,501,402,40101,303,30201,3010101,20202,2020101,
%U A332567 201010101,10101010101,7,601,502,50101,403,40201,4010101,30301,30202,3020101,301010101
%N 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.
%C A332567 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.
%C A332567 The empty partition is encoded as 0.
%H A332567 Alois P. Heinz, <a href="/A332567/b332567.txt">Rows n = 0..34, flattened</a>
%H A332567 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bijective_numeration">Bijective numeration</a>
%H A332567 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A332567 T(6,6) = 30201 encodes the 6th partition of 6: [3,2,1].
%e A332567 T(10,1) = 11 encodes the 1st partition of 10: [10].
%e A332567 T(23,23) = 18040101 encodes the 23rd partition of 23: [17,4,1,1].
%e A332567 Triangle T(n,k) begins:
%e A332567    0;
%e A332567    1;
%e A332567    2, 101;
%e A332567    3, 201, 10101;
%e A332567    4, 301, 202, 20101, 1010101;
%e A332567    5, 401, 302, 30101, 20201, 2010101, 101010101;
%e A332567    6, 501, 402, 40101, 303, 30201, 3010101, 20202, 2020101, ...
%e A332567    7, 601, 502, 50101, 403, 40201, 4010101, 30301, 30202, ...
%e A332567    8, 701, 602, 60101, 503, 50201, 5010101, 404, 40301, 40202, ...
%e A332567    9, 801, 702, 70101, 603, 60201, 6010101, 504, 50301, 50202, ...
%e A332567   11, 901, 802, 80101, 703, 70201, 7010101, 604, 60301, 60202, ...
%e A332567   ...
%p A332567 g:= proc(n) option remember; local d, m, l; m, l:= n, "";
%p A332567       while m>0 do d:= irem(m, 9, 'm');
%p A332567         if d=0 then d:=9; m:= m-1 fi; l:= d, l
%p A332567       od; parse(cat(l))
%p A332567     end:
%p A332567 b:= (n, i)-> `if`(n=0, [""], `if`(i<1, [], [map(x-> cat(
%p A332567      0, g(i), x), b(n-i, min(n-i, i)))[], b(n, i-1)[]])):
%p A332567 T:= n-> map(x-> parse(cat(0, x)), b(n$2))[]:
%p A332567 seq(T(n), n=0..10);
%Y A332567 Column k=1 gives A052382 (for n>0).
%Y A332567 Last row elements give A094028(n-1) (for n>0).
%Y A332567 Cf. A000041, A063008, A080577, A129129, A322761.
%K A332567 nonn,tabf,look,base
%O A332567 0,3
%A A332567 _Alois P. Heinz_, Feb 16 2020