Sequences enumerating triangles of integer partitions Define three conditions P, Q, and R on finite sequences of positive integers y as follows. - y is of type P if y is an integer partition, i.e. y_i >= y_j for i < j. - y is of type Q if y is a strict integer partition, i.e. y_i > y_j for i < j. - y is of type R if y is a constant integer partition, i.e. y_i = y_j for i < j. A triangle or twice-partition of type (C1,C2,C3) and weight n>=1 is a finite sequence of finite sequences of positive integers q=(q_1..q_k) satisfying: - the reversed sorted concatenation of q is of type C1, - the sequence of sums (Sum(q_1)..Sum(q_k)) is of type C2, - each entry q_i is of type C3, and - Sum_{i=1..k} Sum(q_i) = n. Let A(C1,C2,C3)(n) be the number of triangles of type (C1,C2,C3) and weight n. We have the following 27 possibilities. A(PPP) = A063834 = 1 3 6 15 28 66 122 266 503 1027 1913 3874 ... A(PPQ) = A270995 = 1 2 4 7 12 23 37 64 108 180 290 488 ... A(PPR) = A279784 = 1 3 5 12 18 40 60 121 186 344 524 955 ... A(PQP) = A271619 = 1 2 5 8 18 34 65 109 223 386 698 1241 ... A(PQQ) = A279785 = 1 1 3 4 7 11 18 28 47 71 108 166 ... A(PQR) = A279786 = 1 2 4 5 9 16 22 28 49 69 94 138 ... A(PRP) = A279787 = 1 3 4 10 8 29 16 64 58 124 57 469 ... A(PRQ) = A279788 = 1 2 3 4 4 10 6 12 17 21 13 57 ... A(PRR) = A279789 = 1 3 3 8 3 17 3 30 12 41 3 130 ... A(QPP) = A279790 = 1 1 3 3 5 11 12 18 24 49 53 82 ... A(QPQ) = A279790 = 1 1 3 3 5 11 12 18 24 49 53 82 ... A(QPR) = A000009 = 1 1 2 2 3 4 5 6 8 10 12 15 ... A(QQP) = A279375 = 1 1 3 3 5 9 12 16 24 39 49 70 ... A(QQQ) = A279375 = 1 1 3 3 5 9 12 16 24 39 49 70 ... A(QQR) = A000009 = 1 1 2 2 3 4 5 6 8 10 12 15 ... A(QRP) = A279791 = 1 1 2 2 3 6 5 8 8 16 12 23 ... A(QRQ) = A279791 = 1 1 2 2 3 6 5 8 8 16 12 23 ... A(QRR) = A000012 = 1 1 1 1 1 1 1 1 1 1 1 1 ... A(RPP) = A047968 = 1 3 4 8 8 17 16 30 34 52 57 99 ... A(RPQ) = A000005 = 1 2 2 3 2 4 2 4 3 4 2 6 ... A(RPR) = A047968 = 1 3 4 8 8 17 16 30 34 52 57 99 ... A(RQP) = A047966 = 1 2 3 4 4 8 6 10 11 15 13 25 ... A(RQQ) = A000012 = 1 1 1 1 1 1 1 1 1 1 1 1 ... A(RQR) = A047966 = 1 2 3 4 4 8 6 10 11 15 13 25 ... A(RRP) = A007425 = 1 3 3 6 3 9 3 10 6 9 3 18 ... A(RRQ) = A000005 = 1 2 2 3 2 4 2 4 3 4 2 6 ... A(RRR) = A007425 = 1 3 3 6 3 9 3 10 6 9 3 18 ... With duplicates removed there are a total of 18 different sequences: A000005, A000009, A000012, A007425, A047966, A047968, A063834, A270995, A271619, A279375, A279784, A279785, A279786, A279787, A279788, A279789, A279790, A279791.