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A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.

%I A193173 #35 May 22 2020 16:20:42
%S A193173 1,2,1,3,2,1,4,3,2,2,1,5,4,3,3,2,2,1,6,5,4,4,3,3,2,3,2,2,1,7,6,5,5,4,
%T A193173 4,3,4,3,3,2,3,2,2,1,8,7,6,6,5,5,4,5,4,4,3,4,3,3,2,4,3,3,2,2,2,1,9,8,
%U A193173 7,7,6,6,5,6,5,5,4,5,4,4,3,5,4,4,3,3,3,2,4,3,3,2,3,2,2,1,10,9,8,8,7,7,6,7,6
%N A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.
%C A193173 This sequence first differs from A049085 in the partitions of 6 (at flattened index 22):
%C A193173 6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1 (this sequence);
%C A193173 6, 5, 4, 3, 4, 3, 2, 3, 2, 2, 1 (A049085).
%C A193173 - _Jason Kimberley_, Oct 27 2011
%C A193173 Rows sums give A006128, n >= 1. - _Omar E. Pol_, Dec 06 2011
%C A193173 The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A049085.
%H A193173 Alois P. Heinz, <a href="/A193173/b193173.txt">Rows n = 1..26, flattened</a>
%e A193173 The lexicographically ordered partitions of 3 are [[1, 1, 1], [1, 2], [3]], thus row 3 has 3, 2, 1.
%e A193173 Triangle begins:
%e A193173   1;
%e A193173   2, 1;
%e A193173   3, 2, 1;
%e A193173   4, 3, 2, 2, 1;
%e A193173   5, 4, 3, 3, 2, 2, 1;
%e A193173   6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1;
%e A193173   ...
%p A193173 T:= proc(n) local b, ll;
%p A193173       b:= proc(n,l)
%p A193173             if n=0 then ll:= ll, nops(l)
%p A193173             else seq(b(n-i, [l[], i]), i=`if`(l=[], 1, l[-1])..n) fi
%p A193173           end;
%p A193173       ll:= NULL; b(n, []); ll
%p A193173     end:
%p A193173 seq(T(n), n=1..11);
%t A193173 lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t A193173 Table[Length/@Sort[Reverse/@IntegerPartitions[n],lexsort],{n,0,10}] (* _Gus Wiseman_, May 22 2020 *)
%Y A193173 Row lengths are A000041.
%Y A193173 Partition lengths of A026791.
%Y A193173 The version ignoring length is A036043.
%Y A193173 The version for non-reversed partitions is A049085.
%Y A193173 The maxima of these partitions are A194546.
%Y A193173 Reversed partitions in Abramowitz-Stegun order are A036036.
%Y A193173 Reverse-lexicographically ordered partitions are A080577.
%Y A193173 Cf. A001222, A115623, A129129, A185974, A193073, A211992, A228531, A334302, A334434, A334437, A334440, A334441.
%K A193173 nonn,look,tabf
%O A193173 1,2
%A A193173 _Alois P. Heinz_, Jul 17 2011