A236970 -id:A236970 - cp's OEIS frontend

A236971 Number of partitions of n into at least 4 parts from which we can form every partition of n into 4 parts by summing elements.

0, 0, 0, 1, 2, 2, 3, 3, 6, 7, 8, 11, 19, 21, 26, 31, 52, 66, 76, 88, 134, 169, 215, 251, 358, 412, 517, 639, 899, 1065, 1242, 1496, 2072, 2482, 2930, 3449, 4677, 5566

Offset: 1

Jack W Grahl, Feb 02 2014

The corresponding partitions with 2 in the definition instead of 3 are the complete partitions, which A126796 counts.
The qualifier 'into at least 4 parts' is only relevant for n = 1, 2 or 3. It is included because otherwise the condition would be vacuously true for all partitions of 1, 2 and 3. It seems neater to consider that there are no partitions of 1, 2 or 3 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?

			The valid partitions of 7 are (2, 2, 1, 1, 1), (2, 1, 1, 1, 1, 1) and (1, 1, 1, 1, 1, 1, 1). Given any partition of 7 into 4 parts, we can express these four parts as disjoint sums of elements from these partitions. For the third one this is trivial, for the second one because one element of the partition must be at least 2, for the third because in fact two elements of the partition must be at least 2. So a(7) = 3.

A000041 counts partitions, A126796 counts complete partitions - the case for partitions into 2 instead of 4, A236970 and A236972 are the cases for 3 and 5 respectively.

a(30)-a(38) from Willy Van den Driessche, Oct 22 2019

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.

Original entry on oeis.org

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

Offset: 1

Jack W Grahl, Feb 02 2014

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?

			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.

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.

cp's OEIS Frontend

A236971 Number of partitions of n into at least 4 parts from which we can form every partition of n into 4 parts by summing elements.

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