A236972 -id:A236972 - cp's OEIS frontend

A236970 The number of partitions of n into at least 3 parts from which we can form every partition of n into 3 parts by summing elements.

0, 0, 1, 2, 2, 3, 5, 6, 7, 13, 16, 19, 29, 38, 49, 72, 84, 108, 155, 195, 234, 331, 410, 501, 672, 824, 1006, 1341, 1621, 1981, 2583, 3111, 3740, 4846, 5819, 6957, 8787, 10582, 12606, 15840, 18762, 22386, 27851, 32934, 38824, 47961, 56633, 66577, 81168, 95612

Offset: 1

Jack W Grahl, Feb 02 2014

nonn

The corresponding partitions with 2 in the definition instead of 3 are the complete partitions, which are counted by A126796.

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

			The valid partitions of 5 are (2,1,1,1) and (1,1,1,1,1). Given any partition of 5 into 3 parts, it contains one part of at least 2. Therefore we can make any partition of 5 into 3 parts by joining (2,1,1,1) into three sums. (3, 1, 1) is not a valid partition, since (2,2,1) is a partition of 5 into 3 parts which cannot be made by summing elements from (3,1,1). Therefore a(5) = 2.

Jack W Grahl, Haskell code for generating this sequence

Cf. A000041. A126796 is the case for 2 instead of 3, A236971 and A236972 are the cases for 4 and 5.

ric[p_, {x_,y_}] := If[x==0, If[y > Total[p], False, y==0 || AnyTrue[ Reverse@ Union[p], y>=# && ric[ DeleteCases[p, #, 1, 1], {0, y-#}] &]], If[x >= Total[p], False, AnyTrue[ Reverse@ Union@ p, x>=# && ric[ DeleteCases[p, #, 1, 1], {x-#, y}] &]]]; chk[p_] := AllTrue[ Rest /@ IntegerPartitions[Plus @@ p, {3}], ric[p,#] &]; a[n_] := Length@ Select[ IntegerPartitions[n, {3, Infinity}], chk]; Array[a, 24] (* Giovanni Resta, Jul 18 2018 *)

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A236970 The number of partitions of n into at least 3 parts from which we can form every partition of n into 3 parts by summing elements.

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