A085360 -id:A085360 - cp's OEIS frontend

A092078 Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.

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

Offset: 1

Wolfdieter Lang, Mar 19 2004

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).

			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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 36 rows and more comments.

Cf. A092079.

A092079 Characteristic array marking partitions of m whose parts are exponents of partitions of n into m parts.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 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, 1, 0, 1, 1, 0, 1, 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

Offset: 1

Wolfdieter Lang, Mar 19 2004

With N=A000217(n-1) + m, where A000217(n-1) is the largest triangular number less than N, a(N,k)=1 if there is at least one partition of n into m parts which has the parts of the k-th partition of m (in Abramowitz-Stegun order) as exponents. Otherwise a(N,k)=0.
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).

			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=3)=0: There is no partition of 5 into 3 parts which has as exponents 1,1,1, the parts of the third (k=3) partition of 3.
a(N=13,k=2)=1, n=5, m=3; there is a partition of 5 into 3 parts, which has the parts of the second (k=2) partitions of 3, i.e. 1,2, as exponents. In fact there are two such partitions, namely [1^2, 3^1] and [1^1, 2^2].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 36 rows and more comments.

Cf. A092078 (with multiplicities).

cp's OEIS Frontend

A092078 Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs