|
%I #21 Feb 14 2024 16:51:43
%S 1,0,1,0,1,1,0,1,2,1,0,1,4,4,1,0,1,6,12,7,1,0,1,10,31,33,11,1,0,1,14,
%T 73,130,77,16,1,0,1,21,165,464,438,157,22,1,0,1,29,357,1558,2216,1223,
%U 289,29,1,0,1,41,760,5039,10423,8331,2957,492,37,1
%N Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A262495/b262495.txt">Rows n = 0..140, flattened</a>
%e T(3,1) = 1: 3a.
%e T(3,2) = 2: 2a1b, 1a1b1a.
%e T(3,3) = 1: 1a1b1c.
%e T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 1;
%e 0, 1, 4, 4, 1;
%e 0, 1, 6, 12, 7, 1;
%e 0, 1, 10, 31, 33, 11, 1;
%e 0, 1, 14, 73, 130, 77, 16, 1;
%e 0, 1, 21, 165, 464, 438, 157, 22, 1;
%e 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1;
%e 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
%e ...
%p b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
%p b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
%p end:
%p A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
%p T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 24 2017, translated from Maple *)
%Y Columns k=0-10 give: A000007, A057427, A000065, A262445/6 = A320545, A320546, A320547, A320548, A320549, A320550, A320551, A320552.
%Y Main diagonal and lower diagonal give: A000012, A000124 (shifted).
%Y Row sums give A262496.
%Y T(2n,n) gives A262529.
%Y Cf. A256130.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Sep 24 2015
|
