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A344649
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Triangle read by rows where T(n,k) is the number of strict integer partitions of 2n with reverse-alternating sum 2k.
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12
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1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 3, 3, 2, 1, 0, 1, 0, 1, 4, 4, 3, 2, 1, 0, 1, 0, 1, 5, 6, 4, 3, 2, 1, 0, 1, 0, 1, 7, 7, 6, 4, 3, 2, 1, 0, 1, 0, 1, 8, 10, 8, 6, 4, 3, 2, 1, 0, 1
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OFFSET
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0,18
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COMMENTS
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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 T(n,k) is the number of strict integer partitions of 2n into an odd number of parts whose conjugate has exactly 2k odd parts.
Also the number of reversed strict integer partitions of 2n with alternating sum 2k.
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 1 1 0 1
0 1 2 1 0 1
0 1 3 2 1 0 1
0 1 3 3 2 1 0 1
0 1 4 4 3 2 1 0 1
0 1 5 6 4 3 2 1 0 1
0 1 7 7 6 4 3 2 1 0 1
0 1 8 10 8 6 4 3 2 1 0 1
0 1 10 13 12 8 6 4 3 2 1 0 1
0 1 11 18 15 12 8 6 4 3 2 1 0 1
0 1 14 22 21 16 12 8 6 4 3 2 1 0 1
0 1 15 29 27 23 16 12 8 6 4 3 2 1 0 1
Row n = 8 counts the following partitions (empty columns indicated by dots):
. (8,7,1) (7,6,3) (7,5,4) (9,4,3) (11,3,2) (13,2,1) . (16)
(8,6,2) (8,5,3) (10,4,2) (12,3,1)
(9,6,1) (9,5,2) (11,4,1)
(6,4,3,2,1) (10,5,1)
Row n = 9 counts the following partitions (empty columns indicated by dots, A..I = 10..18):
. 981 873 765 954 B43 D32 F21 . I
972 864 A53 C42 E31
A71 963 B52 D41
65421 A62 C51
75321 B61
84321
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MATHEMATICA
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sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&sats[#]==k&]], {n, 0, 30, 2}, {k, 0, n, 2}]
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CROSSREFS
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The non-reversed version is A152146.
The non-reversed non-strict version is A239830.
Right half of even-indexed rows of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A067659 counts strict partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sum of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344604 counts wiggly compositions with twins.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A003242, A006330, A027187, A114121, A116406, A239829, A344607, A344608, A344651, A344654.
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KEYWORD
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AUTHOR
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STATUS
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approved
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