|
| |
|
|
A092078
|
|
Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.
|
|
3
|
|
|
|
1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,24
|
|
|
COMMENTS
|
a(N,k) with N=A000217(n-1) + m, where A000217(n-1) is the largest triangular number less than N, is the number of partitions of n into m parts which have the parts of the k-th partition of m (in Abramowitz-Stegun order) as exponents.
The sequence of row lengths of this array is p(m)= A000041(m) (number of partitions of m) and m is determined from N (the row index) as explained above. It is [1,1,2,1,2,3,1,2,3,5,1,2,3,5,7,1,2,3,5,7,11,...]=A092080(N), N>=1.
One can find the (n,m; k) numbers for the p-th entry (p>2) of the sequence as follows: p= a(n-1) + b(m-1) + k, where a(n-1) := A085360(n-1) is the largest number from the numbers A085360 less than p and b(m-1)=A026905(m-1) is the largest number from the numbers A026905 less than p-a(n-1). p=1 belongs to (1,1;1) and p=2 to (2,1;1).
|
|
|
LINKS
|
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
|
|
|
EXAMPLE
|
N=13 = 10 + 3 with 10=A000217(4), hence n=5 and m=3.
N=10 = 6 + 4 with 6=A000217(3), hence n=4 and m=4.
The sequence entry nr. p=16, which is 0, belongs to (n=4,m=3; k=3)
because 16 = 10 + 3 + 3 with 10=A085360(3), hence n=4 and 3=A026905(2),
hence m=3.
a(N=13,k=2)=2, n=5, m=3; there are exactly 2 partitions of 5 into 3 parts, each having the parts of the second (k=2) partition of 3, i.e. 1,2, as exponents. These two 3-partitions of 5 are: [1^2, 3^1] and [1^1, 2^2], which are all the 3-partitions of 5 because the other entries of row N=13 are 0.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
