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Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.
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%I #38 Sep 22 2023 08:58:20

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

%C Row n lists the first A000041(n) terms of A141285.

%C 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 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>

%F a(n) = A061395(A334437(n)). - _Gus Wiseman_, May 31 2020

%e For n = 5 the partitions of 5 in colexicographic order are:

%e 1+1+1+1+1

%e 2+1+1+1

%e 3+1+1

%e 2+2+1

%e 4+1

%e 3+2

%e 5

%e so the fifth row is the largest in each partition: 1,2,3,2,4,3,5

%e Triangle begins:

%e 1;

%e 1,2;

%e 1,2,3;

%e 1,2,3,2,4;

%e 1,2,3,2,4,3,5;

%e 1,2,3,2,4,3,5,2,4,3,6;

%e 1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;

%e 1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;

%e ...

%t colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];

%t Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* _Gus Wiseman_, May 31 2020 *)

%Y The sum of row n is A006128(n).

%Y Cf. A135010, A138121, A141285, A194547, A194548, A194549.

%Y Row lengths are A000041.

%Y Let y be the n-th integer partition in colexicographic order (A211992):

%Y - The maximum of y is a(n).

%Y - The length of y is A193173(n).

%Y - The minimum of y is A196931(n).

%Y - The Heinz number of y is A334437(n).

%Y Lexicographically ordered reversed partitions are A026791.

%Y Reverse-colexicographically ordered partitions are A026792.

%Y Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.

%Y Reverse-lexicographically ordered partitions are A080577.

%Y Lexicographically ordered partitions are A193073.

%Y Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Dec 10 2011

%E Definition corrected by _Omar E. Pol_, Sep 12 2013