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A363493 Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.

%I A363493 #31 Sep 05 2023 09:38:00
%S A363493 1,1,1,1,2,2,1,4,6,4,1,10,18,17,6,1,25,61,68,38,10,1,75,210,292,202,
%T A363493 83,14,1,225,778,1252,1116,576,170,22,1,780,3008,5670,5928,3899,1490,
%U A363493 341,30,1,2704,12219,26114,32382,25320,12655,3856,678,46,1,10556,52268,126073,177666,163695,98282,39230,9418,1319,62,1
%N A363493 Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.
%H A363493 Alois P. Heinz, <a href="/A363493/b363493.txt">Rows n = 0..150, flattened</a>
%H A363493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363493 Sum_{k=0..max(0,n-1)} k * T(n,k) = A363496(n).
%e A363493 T(4,0) = 4: 13|24, 13|2|4, 1|24|3, 1|2|3|4.
%e A363493 T(4,1) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.
%e A363493 T(4,2) = 4: 123|4, 12|34, 14|23, 1|234.
%e A363493 T(4,3) = 1: 1234.
%e A363493 T(5,2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
%e A363493 Triangle T(n,k) begins:
%e A363493      1;
%e A363493      1;
%e A363493      1,     1;
%e A363493      2,     2,     1;
%e A363493      4,     6,     4,     1;
%e A363493     10,    18,    17,     6,     1;
%e A363493     25,    61,    68,    38,    10,     1;
%e A363493     75,   210,   292,   202,    83,    14,    1;
%e A363493    225,   778,  1252,  1116,   576,   170,   22,   1;
%e A363493    780,  3008,  5670,  5928,  3899,  1490,  341,  30,  1;
%e A363493   2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1;
%e A363493   ...
%p A363493 b:= proc(n, x, y) option remember; `if`(n=0, 1,
%p A363493      `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
%p A363493         b(n-1, y, x)*x + b(n-1, y, x+1))
%p A363493     end:
%p A363493 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
%p A363493 seq(T(n), n=0..12);
%t A363493 b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1,
%t A363493   If[y == 0, 0, Expand[b[n - 1, y - 1, x + 1]*y*z]] +
%t A363493   b[n - 1, y, x]*x + b[n - 1, y, x + 1]];
%t A363493 T[n_] := CoefficientList[b[n, 0, 0], z];
%t A363493 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Sep 05 2023, after _Alois P. Heinz_ *)
%Y A363493 Columns k=0-2 give: A124419, A363511, A363588.
%Y A363493 Row sums give A000110.
%Y A363493 T(n+1,n) gives A000012.
%Y A363493 T(n+2,n) gives A027383.
%Y A363493 T(2n+1,n) gives A363495.
%Y A363493 Cf. A152874, A363496, A363519.
%K A363493 nonn,tabf
%O A363493 0,5
%A A363493 _Alois P. Heinz_, Jun 05 2023