A236972 The number of partitions of n into at least 5 parts from which we can form every partition of n into 5 parts by summing elements.

0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 7, 8, 9, 13, 14, 24, 29, 35, 38

Offset: 1

Author

Jack W Grahl, Feb 02 2014

Keywords

• nonn
• more

Comments

The corresponding partitions with 2 in the definition instead of 5 are the complete partitions, which A126796 counts.

The qualifier 'into at least 5 parts' is only relevant for n = 1, 2, 3 or 4. It is included because otherwise the condition would be vacuously true for all partitions of 1, 2, 3 and 4. It seems neater to consider that there are no partitions of 1, 2, 3 or 4 of this form.

What is the limit for large n of the proportion of partitions of n for which this holds, or this sequence divided by A000041?

Examples

			The valid partitions of 11 are all those which contain only 1's, 2's and 3's, with no more than one 3 and no more than three 2's or 3's. This is because every partition of 11 into 5 parts contains at least one element 3 or more, and at least 3 elements 2 or more. There are 7 such partitions, therefore a(11) = 7.
		

Crossrefs

A000041 counts partitions, A126796 counts complete partitions - the case for partitions into 2 instead of 5, A236970 and A236971 are the cases for 3 and 4 respectively.