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A225597
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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.
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1
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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
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OFFSET
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1,3
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COMMENTS
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For the definition of "region" see A206437.
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).
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LINKS
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EXAMPLE
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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.
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. Diagram Illustration of parts ending in column k:
. for n=5 k=1 k=2 k=3 k=4 k=5
. _ _ _ _ _ _ _ _ _ _
. |_ _ _ | _ _ _ |_ _ _ _ _|
. |_ _ _|_ | |_ _ _| _ _ _ _ |_ _|
. |_ _ | | _ _ |_ _ _ _| |_|
. |_ _|_ | | |_ _| _ _ _ |_ _| |_|
. |_ _ | | | _ _ |_ _ _| |_| |_|
. |_ | | | | _ |_ _| |_| |_| |_|
. |_|_|_|_|_| |_| |_| |_| |_| |_|
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k = 1 2 3 4 5
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The 5th row lists: 1 3 4 5 7
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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;
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CROSSREFS
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Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446, A206437, A207779, A211978, A220517, A225598, A225600, A225610.
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KEYWORD
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AUTHOR
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STATUS
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approved
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