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Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.
27

%I #13 Mar 11 2023 23:06:27

%S 1,1,0,1,1,0,1,1,1,0,1,0,3,1,0,1,0,2,3,1,0,1,0,1,4,4,1,0,1,0,0,3,6,4,

%T 1,0,1,0,0,1,7,7,5,1,0,1,0,0,1,4,8,10,5,1,0,1,0,0,0,3,6,14,11,6,1,0,1,

%U 0,0,0,1,5,12,16,14,6,1,0

%N Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

%C Also the number of integer partitions of n whose right half (inclusive) sums to n-k.

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 1 1 1 0

%e 1 0 3 1 0

%e 1 0 2 3 1 0

%e 1 0 1 4 4 1 0

%e 1 0 0 3 6 4 1 0

%e 1 0 0 1 7 7 5 1 0

%e 1 0 0 1 4 8 10 5 1 0

%e 1 0 0 0 3 6 14 11 6 1 0

%e 1 0 0 0 1 5 12 16 14 6 1 0

%e 1 0 0 0 1 2 12 14 23 16 7 1 0

%e 1 0 0 0 0 2 7 13 24 27 19 7 1 0

%e 1 0 0 0 0 1 5 9 24 30 35 21 8 1 0

%e 1 0 0 0 0 1 3 7 17 31 42 40 25 8 1 0

%e 1 0 0 0 0 0 2 4 16 23 46 51 51 27 9 1 0

%e 1 0 0 0 0 0 1 3 10 21 37 57 69 57 31 9 1 0

%e 1 0 0 0 0 0 1 2 7 15 34 47 83 81 69 34 10 1 0

%e For example, row n = 9 counts the following partitions:

%e (9) . . (333) (432) (54) (63) (72) (81)

%e (441) (522) (621) (711)

%e (22221) (531) (3321) (4311)

%e (111111111) (3222) (4221) (5211)

%e (32211) (33111) (6111)

%e (2211111) (42111)

%e (3111111) (51111)

%e (21111111) (222111)

%e (321111)

%e (411111)

%e For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).

%t Table[Length[Select[IntegerPartitions[n], Total[Take[#,Floor[Length[#]/2]]]==k&]],{n,0,10},{k,0,n}]

%Y Row sums are A000041.

%Y Column sums are A360673, inclusive A360671.

%Y The central diagonal T(2n,n) is A360674, ranks A360953.

%Y The left inclusive version is A360675 with rows reversed.

%Y A008284 counts partitions by length.

%Y A359893 and A359901 count partitions by median.

%Y First for prime indices, second for partitions, third for prime factors:

%Y - A360676 gives left sum (exclusive), counted by A360672, product A361200.

%Y - A360677 gives right sum (exclusive), counted by A360675, product A361201.

%Y - A360678 gives left sum (inclusive), counted by A360675, product A347043.

%Y - A360679 gives right sum (inclusive), counted by A360672, product A347044.

%Y Cf. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616, A360682, A360686.

%K nonn,tabl

%O 0,13

%A _Gus Wiseman_, Feb 27 2023