cp's OEIS Frontend

A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

%I A357705 #6 Oct 10 2022 20:47:08
%S A357705 1,0,1,0,1,1,0,2,0,1,0,2,2,0,1,0,3,1,2,0,1,0,3,2,3,2,0,1,0,4,2,4,1,3,
%T A357705 0,1,0,4,3,3,6,2,3,0,1,0,5,3,5,3,7,2,4,0,1,0,5,4,5,4,9,7,3,4,0,1,0,6,
%U A357705 4,7,3,12,5,10,3,5,0,1
%N A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
%C A357705 We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...
%e A357705 Triangle begins:
%e A357705   1
%e A357705   0  1
%e A357705   0  1  1
%e A357705   0  2  0  1
%e A357705   0  2  2  0  1
%e A357705   0  3  1  2  0  1
%e A357705   0  3  2  3  2  0  1
%e A357705   0  4  2  4  1  3  0  1
%e A357705   0  4  3  3  6  2  3  0  1
%e A357705   0  5  3  5  3  7  2  4  0  1
%e A357705   0  5  4  5  4  9  7  3  4  0  1
%e A357705   0  6  4  7  3 12  5 10  3  5  0  1
%e A357705   0  6  5  7  5 10 16  7 11  4  5  0  1
%e A357705   0  7  5  9  5 14 11 18  7 14  4  6  0  1
%e A357705 Row n = 7 counts the following reversed partitions:
%e A357705   .  (16)   (25)   (34)       (1123)  (1114)   .  (7)
%e A357705      (115)  (223)  (1222)             (11113)
%e A357705      (124)         (111112)           (11122)
%e A357705      (133)         (1111111)
%t A357705 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357705 Table[Length[Select[Reverse/@IntegerPartitions[n],skats[#]==k&]],{n,0,11},{k,-n,n,2}]
%Y A357705 Row sums are A000041.
%Y A357705 First nonzero entry of each row is A004526.
%Y A357705 The central column is A357640, half A357639.
%Y A357705 For original alternating sum we have A344651, ordered A097805.
%Y A357705 The half-alternating version is A357704.
%Y A357705 The ordered non-reverse version (compositions) is A357646, half A357645.
%Y A357705 The non-reverse version is A357638, half A357637.
%Y A357705 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357705 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357705 A357621 gives half-alternating sum of standard compositions, skew A357623.
%Y A357705 A357629 gives half-alternating sum of prime indices, skew A357630.
%Y A357705 A357633 gives half-alternating sum of Heinz partition, skew  A357634.
%Y A357705 Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.
%K A357705 nonn,tabl
%O A357705 0,8
%A A357705 _Gus Wiseman_, Oct 10 2022