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Number of even-length integer partitions of n into parts that are alternately unequal and equal.
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These are partitions whose run-lengths multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.
Not requiring the Without equalities gives we have A351008, any length A122129, opposite A122135.
The non-strict version is Without inequalities we have A351012, any length A351003, opposite A351004.
Cf. A000070, A003242, A018819, A027383, `A035363, `A087897, `A088218, `A101417, A122134, `A344605, ~A345194, ~A350837, ~A350839, A350842, A350844, `A351011.
allocated for Gus WisemanNumber of even-length integer partitions of n that are alternately unequal and equal.
1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807
0,6
These are partitions whose run-lengths begin with 1, are followed by any number of 2's, and end with another 1.
The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
21 31 32 42 43 53 54 64 65 75 76 86 87
41 51 52 62 63 73 74 84 85 95 96
61 71 72 82 83 93 94 A4 A5
3221 81 91 92 A2 A3 B3 B4
4221 5221 A1 B1 B2 C2 C3
4331 4332 C1 D1 D2
6221 5331 5332 5441 E1
7221 6331 6332 5442
8221 7331 6441
9221 7332
8331
A221
433221
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&&And@@Table[#[[i]]!=#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]
The alternately equal and unequal version is A035457, any length A351005.
This is the even-length case of A351006, odd-length A053251.
Not requiring the equalities gives A351008, any length A122129, opposite A122135.
The non-strict version is A351012, any length A351003, opposite A351004.
Cf. A000070, A003242, A018819, A027383, `A035363, `A087897, `A088218, `A101417, A122134, `A344605, ~A345194, ~A350837, ~A350839, A350842, A350844, `A351011.
allocated
nonn
Gus Wiseman, Jan 31 2022
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allocated for Gus Wiseman
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