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A343941
Number of strict integer partitions of 2n with reverse-alternating sum 4.
10
0, 0, 1, 0, 1, 2, 3, 3, 4, 5, 7, 8, 10, 11, 14, 15, 18, 20, 23, 25, 29, 31, 35, 38, 42, 45, 50, 53, 58, 62, 67, 71, 77, 81, 87, 92, 98, 103, 110, 115, 122, 128, 135, 141, 149, 155, 163, 170, 178, 185, 194, 201, 210, 218, 227, 235, 245, 253, 263, 272, 282, 291, 302
OFFSET
0,6
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts, so a(n) is the number of strict odd-length integer partitions of 2n whose conjugate has exactly 4 odd parts (first example). By conjugation, this is also the number partitions of 2n covering an initial interval and containing exactly four odd parts, one of which is the greatest (second example).
EXAMPLE
The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
4 . 521 532 543 653 763 873 983 A93 BA3
631 642 752 862 972 A82 B92 CA2
741 851 961 A71 B81 C91 DA1
64321 65421 65432 76432 76542
75321 75431 76531 86541
76421 86431 87432
86321 87421 87531
97321 97431
98421
A8321
The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
1111 . 32111 33211 33321 333221 543211 543321
322111 332211 3322211 3332221 5432211
3222111 32222111 33222211 33322221
322222111 332222211
3222222111
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&sats[#]==4&]], {n, 0, 30, 2}]
CROSSREFS
The non-reverse non-strict version is A000710.
The non-reverse version is A026810.
The non-strict version is column k = 2 of A344610.
This is column k = 2 of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Sequence in context: A367220 A141472 A029034 * A280127 A237977 A115339
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 09 2021
EXTENSIONS
More terms from Bert Dobbelaere, Jun 12 2021
STATUS
approved