%I #28 Nov 17 2023 11:21:10

%S 1,1,0,2,0,1,4,0,3,4,8,0,2,18,14,18,0,7,27,87,42,40,0,5,102,162,360,

%T 147,101,0,20,179,866,931,1456,434,254,0,15,675,1746,5836,4755,5778,

%U 1619,723,0,67,1321,9087,16416,36031,22893,23052,5044,2064,0,52,5216,19863,93452,117172,206570,115178,94210,20271,6586

%N Number T(n,k) of partitions of [n] having exactly k parity changes within the partition, n>=0, 0<=k<=max(0,n-1), read by rows.

%C The blocks are ordered with increasing least elements.

%H Alois P. Heinz, <a href="/A363519/b363519.txt">Rows n = 0..27, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F Sum_{k=0..max(0,n-1)} k * T(n,k) = A363549(n).

%e T(4,1) = 3: 134|2, 13|24, 13|2|4.

%e T(4,2) = 4: 124|3, 14|23, 14|2|3, 1|24|3.

%e T(4,3) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.

%e T(5,2) = 18: 1245|3, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5.

%e T(5,4) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 0, 2;

%e 0, 1, 4;

%e 0, 3, 4, 8;

%e 0, 2, 18, 14, 18;

%e 0, 7, 27, 87, 42, 40;

%e 0, 5, 102, 162, 360, 147, 101;

%e 0, 20, 179, 866, 931, 1456, 434, 254;

%e 0, 15, 675, 1746, 5836, 4755, 5778, 1619, 723;

%e 0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064;

%e ...

%p b:= proc(l, i, t) option remember; expand(`if`(l=[], 1,

%p add((f-> b(subsop(j=[][], l), j, `if`(f, 1-t, t))*

%p `if`(f, x, 1))(l[j]=t), j=[1, $i..nops(l)])))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(

%p b([ seq(irem(i, 2), i=2..n)], 1, 0)):

%p seq(T(n), n=0..12);

%t b[l_, i_, t_] := b[l, i, t] = Expand[If[l == {}, 1, Sum[Function[f, b[ReplacePart[l, j -> Nothing], j, If[f, 1 - t, t]]*If[f, x, 1]][l[[j]] == t], {j, Join[{1}, Range[i, Length@l]]}]]];

%t T[n_] := CoefficientList[b[ Table[Mod[i, 2], {i, 2, n}], 1, 0], x];

%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 17 2023, after _Alois P. Heinz_ *)

%Y Column k=1 gives A363550.

%Y Row sums give A000110.

%Y T(n,max(0,n-1)) gives A274547.

%Y Cf. A363493, A363549.

%K nonn,tabf

%O 0,4

%A _Alois P. Heinz_, Jun 07 2023