A329255 - OEIS

%I #24 Nov 25 2019 01:05:07

%S 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 12,10,13,13,12,10,14,13,12,14,15,15,15,14,15,16,15,15,14,15,17,17,15,

%U 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 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 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 Also this triangle can be constructed replacing every zero of triangle A285891 with the previous positive integer from the same column.

%e Triangle begins:

%e    1;

%e    2;

%e    3,  3;

%e    4,  3;

%e    5,  5;

%e    6,  5,  6;

%e    7,  7,  6;

%e    8,  7,  6;

%e    9,  9,  9;

%e   10,  9,  9, 10;

%e   11, 11,  9, 10;

%e   12, 11, 12, 10;

%e   13, 13, 12, 10;

%e   14, 13, 12, 14;

%e   15, 15, 15, 14, 15;

%e   16, 15, 15, 14, 15;

%e   17, 17, 15, 14, 15;

%e   18, 17, 18, 18, 15;

%e   19, 19, 18, 18, 15;

%e   20, 19, 18, 18, 20;

%e   21, 21, 21, 18, 20, 21;

%e   22, 21, 21, 22, 20, 21;

%e   23, 23, 21, 22, 20, 21;

%e   24, 23, 24, 22, 20, 21;

%e   25, 25, 24, 22, 25, 21;

%e   26, 25, 24, 26, 25, 21;

%e   27, 27, 27, 26, 25, 27;

%e   28, 27, 27, 26, 25, 27, 28;

%e ...

%Y Column k stars with A000217(k) in the row A000217(k).

%Y Row n has length A003056(n).

%Y Cf. A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Nov 09 2019