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A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.

%I A329255 #24 Nov 25 2019 01:05:07
%S A329255 1,2,3,3,4,3,5,5,6,5,6,7,7,6,8,7,6,9,9,9,10,9,9,10,11,11,9,10,12,11,
%T A329255 12,10,13,13,12,10,14,13,12,14,15,15,15,14,15,16,15,15,14,15,17,17,15,
%U A329255 14,15,18,17,18,18,15,19,19,18,18,15,20,19,18,18,20,21,21,21,18,20,21,22,21,21,22,20,21
%N A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.
%C A329255 T(n,k) is also the positive integer whose partition into k consecutive parts is associated to the k-th vertex, from left to right, of the largest Dyck path of the symmetric representation of sigma(n). For more information see A237593.
%C A329255 Also this triangle can be constructed replacing every zero of triangle A285891 with the previous positive integer from the same column.
%e A329255 Triangle begins:
%e A329255    1;
%e A329255    2;
%e A329255    3,  3;
%e A329255    4,  3;
%e A329255    5,  5;
%e A329255    6,  5,  6;
%e A329255    7,  7,  6;
%e A329255    8,  7,  6;
%e A329255    9,  9,  9;
%e A329255   10,  9,  9, 10;
%e A329255   11, 11,  9, 10;
%e A329255   12, 11, 12, 10;
%e A329255   13, 13, 12, 10;
%e A329255   14, 13, 12, 14;
%e A329255   15, 15, 15, 14, 15;
%e A329255   16, 15, 15, 14, 15;
%e A329255   17, 17, 15, 14, 15;
%e A329255   18, 17, 18, 18, 15;
%e A329255   19, 19, 18, 18, 15;
%e A329255   20, 19, 18, 18, 20;
%e A329255   21, 21, 21, 18, 20, 21;
%e A329255   22, 21, 21, 22, 20, 21;
%e A329255   23, 23, 21, 22, 20, 21;
%e A329255   24, 23, 24, 22, 20, 21;
%e A329255   25, 25, 24, 22, 25, 21;
%e A329255   26, 25, 24, 26, 25, 21;
%e A329255   27, 27, 27, 26, 25, 27;
%e A329255   28, 27, 27, 26, 25, 27, 28;
%e A329255 ...
%Y A329255 Column k stars with A000217(k) in the row A000217(k).
%Y A329255 Row n has length A003056(n).
%Y A329255 Cf. A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.
%K A329255 nonn,tabf
%O A329255 1,2
%A A329255 _Omar E. Pol_, Nov 09 2019