%I #12 Dec 10 2021 11:11:12

%S 1,1,1,1,2,1,2,2,1,4,2,1,3,5,2,1,7,5,2,1,5,9,5,2,1,12,10,5,2,1,7,17,

%T 10,5,2,1,19,19,10,5,2,1,11,28,20,10,5,2,1,30,33,20,10,5,2,1,15,47,35,

%U 20,10,5,2,1,45,57,36,20,10,5,2,1,22,73,62,36,20,10,5,2,1

%N Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

%C The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

%C Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - _Gus Wiseman_, Nov 29 2021

%e Triangle begins:

%e 1

%e 1

%e 1 1

%e 2 1

%e 2 2 1

%e 4 2 1

%e 3 5 2 1

%e 7 5 2 1

%e 5 9 5 2 1

%e 12 10 5 2 1

%e 7 17 10 5 2 1

%e 19 19 10 5 2 1

%e 11 28 20 10 5 2 1

%e 30 33 20 10 5 2 1

%e 15 47 35 20 10 5 2 1

%e 45 57 36 20 10 5 2 1

%e 22 73 62 36 20 10 5 2 1

%e 67 92 64 36 20 10 5 2 1

%e 30 114 102 65 36 20 10 5 2 1

%e 97 147 107 65 36 20 10 5 2 1

%e Row n = 10 counts the following partitions (A = 10):

%e (55) (64) (73) (82) (91) (A)

%e (3322) (442) (433) (622) (811)

%e (4411) (541) (532) (721)

%e (222211) (3331) (631) (7111)

%e (331111) (4222) (5221) (61111)

%e (22111111) (4321) (6211)

%e (1111111111) (5311) (42211)

%e (22222) (52111)

%e (32221) (511111)

%e (33211) (4111111)

%e (43111)

%e (322111)

%e (421111)

%e (2221111)

%e (3211111)

%e (31111111)

%e (211111111)

%e The conjugate version is:

%e (A) (55) (3331) (331111) (31111111) (1111111111)

%e (64) (73) (5311) (511111) (211111111)

%e (82) (91) (7111) (3211111)

%e (442) (433) (33211) (4111111)

%e (622) (532) (43111) (22111111)

%e (4222) (541) (52111)

%e (22222) (631) (61111)

%e (721) (322111)

%e (811) (421111)

%e (3322) (2221111)

%e (4321)

%e (4411)

%e (5221)

%e (6211)

%e (32221)

%e (42211)

%e (222211)

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Table[Length[Select[IntegerPartitions[n],ats[#]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

%Y This is A103919 with all zeros removed.

%Y The strict version is A152146 interleaved with A152157.

%Y The rows are those of A239830 interleaved with those of A239829.

%Y The reverse version is the right half of A344612.

%Y The strict reverse version is the right half of A344739.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.

%Y A124754 lists alternating sums of standard compositions (reverse: A344618).

%Y A316524 is the alternating sum of the prime indices of n (reverse: A344616).

%Y A325534/A325535 count separable/inseparable partitions.

%Y A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.

%Y A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.

%Y A344610 counts partitions of n by positive rev-alternating sum.

%Y A344611 counts partitions of 2n with rev-alternating sum >= 0.

%Y A345197 counts compositions by sum, length, and alternating sum.

%Y A346697 gives the sum of odd-indexed prime indices (reverse: A346699).

%Y A346702 represents the odd bisection of compositions, sums A209281.

%Y Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.

%K nonn,tabf

%O 0,5

%A _Gus Wiseman_, Jun 05 2021