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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.

%I A322761 #23 Oct 21 2020 03:47:13
%S A322761 1,2,11,3,21,111,4,31,22,211,1111,5,41,32,311,221,2111,11111,6,51,42,
%T A322761 411,33,321,3111,222,2211,21111,111111,7,61,52,511,43,421,4111,331,
%U A322761 322,3211,31111,2221,22111,211111,1111111
%N 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.
%C A322761 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.
%H A322761 Alois P. Heinz, <a href="/A322761/b322761.txt">Rows n = 1..28, flattened</a>
%e A322761 Triangle begins:
%e A322761 1,
%e A322761 2, 11,
%e A322761 3, 21, 111,
%e A322761 4, 31, 22, 211, 1111,
%e A322761 5, 41, 32, 311, 221, 2111, 11111,
%e A322761 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111,
%e A322761 7, 61, 52, 511, 43, 421, 4111, 331, 322, 3211, 31111, 2221, 22111, 211111, 1111111,
%e A322761 ...
%e A322761 ...
%p A322761 b:= (n, i)-> `if`(n=0 or i=1, [cat(1$n)], [map(x->
%p A322761     cat(i, x), b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A322761 T:= n-> map(parse, b(n$2))[]:
%p A322761 seq(T(n), n=1..10);  # _Alois P. Heinz_, Dec 30 2018
%t A322761 revlexsort[f_, c_] := OrderedQ[PadRight[{c, f}]];
%t A322761 Table[FromDigits /@ Sort[IntegerPartitions[n], revlexsort], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Oct 20 2020, after _Gus Wiseman_ in A080577 *)
%Y A322761 Cf. A000041 (number of terms in row n), A036037, A080577.
%Y A322761 See also A006128.
%Y A322761 First column gives A000027.
%Y A322761 Last elements of rows give A000042.
%K A322761 nonn,look,tabf,base,fini
%O A322761 1,2
%A A322761 _N. J. A. Sloane_, Dec 30 2018