cp's OEIS Frontend

A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

%I A225597 #24 Mar 13 2015 22:57:55
%S A225597 1,1,2,1,2,3,1,3,3,5,1,3,4,5,7,1,4,5,7,7,11,1,4,6,8,9,11,15,1,5,7,11,
%T A225597 10,15,15,22,1,5,9,12,13,17,19,22,30,1,6,10,16,15,22,21,29,30,42,1,6,
%U A225597 12,18,19,25,26,32,38,42,56,1,7,14,23,22,33,29,41,42,54,56,77
%N A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.
%C A225597 For the definition of "region" see A206437.
%C A225597 T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).
%e A225597 For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
%e A225597 .
%e A225597 .    Diagram    Illustration of parts ending in column k:
%e A225597 .    for n=5      k=1   k=2     k=3       k=4        k=5
%e A225597 .   _ _ _ _ _                                  _ _ _ _ _
%e A225597 .  |_ _ _    |                _ _ _           |_ _ _ _ _|
%e A225597 .  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
%e A225597 .  |_ _    | |          _ _          |_ _ _ _|        |_|
%e A225597 .  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
%e A225597 .  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
%e A225597 .  |_  | | | |      _  |_ _|     |_|       |_|        |_|
%e A225597 .  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
%e A225597 .
%e A225597 k = 1 2 3 4 5
%e A225597 .
%e A225597 The 5th row lists:  1     3       4         5          7
%e A225597 .
%e A225597 Triangle begins:
%e A225597 1;
%e A225597 1,  2;
%e A225597 1,  2,  3;
%e A225597 1,  3,  3,  5;
%e A225597 1,  3,  4,  5,  7;
%e A225597 1,  4,  5,  7,  7, 11;
%e A225597 1,  4,  6,  8,  9, 11, 15;
%e A225597 1,  5,  7, 11, 10, 15, 15, 22;
%e A225597 1,  5,  9, 12, 13, 17, 19, 22, 30;
%e A225597 1,  6, 10, 16, 15, 22, 21, 29, 30, 42;
%e A225597 1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
%e A225597 1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;
%Y A225597 Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.
%Y A225597 Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446,  A206437, A207779, A211978, A220517, A225598, A225600, A225610.
%K A225597 nonn,tabl
%O A225597 1,3
%A A225597 _Omar E. Pol_, Aug 02 2013