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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

%I A194546 #38 Sep 22 2023 08:58:20
%S A194546 1,1,2,1,2,3,1,2,3,2,4,1,2,3,2,4,3,5,1,2,3,2,4,3,5,2,4,3,6,1,2,3,2,4,
%T A194546 3,5,2,4,3,6,3,5,4,7,1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8,1,2,
%U A194546 3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8,3,5,4,7,3,6,5,9
%N A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.
%C A194546 Row n lists the first A000041(n) terms of A141285.
%C A194546 The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - _Joerg Arndt_, Sep 13 2013
%H A194546 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A194546 a(n) = A061395(A334437(n)). - _Gus Wiseman_, May 31 2020
%e A194546 For n = 5 the partitions of 5 in colexicographic order are:
%e A194546   1+1+1+1+1
%e A194546   2+1+1+1
%e A194546   3+1+1
%e A194546   2+2+1
%e A194546   4+1
%e A194546   3+2
%e A194546   5
%e A194546 so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
%e A194546 Triangle begins:
%e A194546   1;
%e A194546   1,2;
%e A194546   1,2,3;
%e A194546   1,2,3,2,4;
%e A194546   1,2,3,2,4,3,5;
%e A194546   1,2,3,2,4,3,5,2,4,3,6;
%e A194546   1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
%e A194546   1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
%e A194546 ...
%t A194546 colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
%t A194546 Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* _Gus Wiseman_, May 31 2020 *)
%Y A194546 The sum of row n is A006128(n).
%Y A194546 Cf. A135010, A138121, A141285, A194547, A194548, A194549.
%Y A194546 Row lengths are A000041.
%Y A194546 Let y be the n-th integer partition in colexicographic order (A211992):
%Y A194546 - The maximum of y is a(n).
%Y A194546 - The length of y is A193173(n).
%Y A194546 - The minimum of y is A196931(n).
%Y A194546 - The Heinz number of y is A334437(n).
%Y A194546 Lexicographically ordered reversed partitions are A026791.
%Y A194546 Reverse-colexicographically ordered partitions are A026792.
%Y A194546 Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
%Y A194546 Reverse-lexicographically ordered partitions are A080577.
%Y A194546 Lexicographically ordered partitions are A193073.
%Y A194546 Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.
%K A194546 nonn,tabf
%O A194546 1,3
%A A194546 _Omar E. Pol_, Dec 10 2011
%E A194546 Definition corrected by _Omar E. Pol_, Sep 12 2013