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.
Original entry on oeis.org
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, 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, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77
Offset: 1
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.
.
. Diagram Illustration of parts ending in column k:
. for n=5 k=1 k=2 k=3 k=4 k=5
. _ _ _ _ _ _ _ _ _ _
. |_ _ _ | _ _ _ |_ _ _ _ _|
. |_ _ _|_ | |_ _ _| _ _ _ _ |_ _|
. |_ _ | | _ _ |_ _ _ _| |_|
. |_ _|_ | | |_ _| _ _ _ |_ _| |_|
. |_ _ | | | _ _ |_ _ _| |_| |_|
. |_ | | | | _ |_ _| |_| |_| |_|
. |_|_|_|_|_| |_| |_| |_| |_| |_|
.
k = 1 2 3 4 5
.
The 5th row lists: 1 3 4 5 7
.
Triangle begins:
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, 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, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;
Cf.
A006128,
A133041,
A135010,
A138137,
A139582,
A141285,
A182377,
A186114,
A186412,
A187219,
A193870,
A194446,
A206437,
A207779,
A211978,
A220517,
A225598,
A225600,
A225610.
A225599
Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.
Original entry on oeis.org
1, 3, 1, 6, 1, 2, 12, 1, 4, 3, 20, 1, 4, 5, 5, 35, 1, 6, 8, 9, 7, 54, 1, 6, 10, 12, 11, 11, 86, 1, 8, 13, 20, 14, 19, 15, 128, 1, 8, 18, 23, 22, 25, 23, 22, 192, 1, 10, 21, 34, 30, 37, 29, 36, 30, 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42, 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56
Offset: 1
For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.
.
. Illustration of the parts
. Diagram that start in column k:
. for n=5 k=1 k=2 k=3 k=4 k=5
. _ _ _ _ _ _ _ _ _ _
. |_ _ _ | |_ _ _ _ _| _ _
. |_ _ _|_ | |_ _ _|_ |_ _| _
. |_ _ | | |_ _ _ _| _ _ |_|
. |_ _|_ | | |_ _|_ |_ _| _ |_|
. |_ _ | | | |_ _ _| _ |_| |_|
. |_ | | | | |_ _| _ |_| |_| |_|
. |_|_|_|_|_| |_| |_| |_| |_| |_|
.
k = 1 2 3 4 5
.
The 5th row lists: 20 1 4 5 5
.
Triangle begins:
1;
3, 1;
6, 1, 2;
12, 1, 4, 3;
20, 1, 4, 5, 5;
35, 1, 6, 8, 9, 7;
54, 1, 6, 10, 12, 11, 11;
86, 1, 8, 13, 20, 14, 19, 15;
128, 1, 8, 18, 23, 22, 25, 23, 22;
192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;
Cf.
A135010,
A141285,
A176572,
A186114,
A186412,
A187219,
A193870,
A194446,
A206437,
A211978,
A225598,
A225600,
A225610.
Showing 1-2 of 2 results.
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