A287215 - OEIS

%I #25 Dec 29 2018 09:36:51

%S 1,1,1,1,1,3,1,1,8,5,1,1,22,21,7,1,1,65,86,39,11,1,1,209,361,209,77,

%T 19,1,1,732,1584,1123,493,171,35,1,1,2780,7315,6153,3124,1293,413,67,

%U 1,1,11377,35635,34723,20019,9320,3709,1059,131,1,1,49863,183080,202852,130916,66992,30396,11373,2837,259,1

%N Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

%C The maximal absolute difference is assumed to be zero if there are fewer than two blocks.

%C T(n,k) is defined for all n,k >= 0.  The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

%H Alois P. Heinz, <a href="/A287215/b287215.txt">Rows n = 0..141, flattened</a>

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

%F T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1.

%e T(4,0) = 1: 1234.

%e T(4,1) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

%e T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.

%e T(4,3) = 1: 123|4.

%e Triangle T(n,k) begins:

%e   1;

%e   1;

%e   1,   1;

%e   1,   3,    1;

%e   1,   8,    5,    1;

%e   1,  22,   21,    7,   1;

%e   1,  65,   86,   39,  11,   1;

%e   1, 209,  361,  209,  77,  19,  1;

%e   1, 732, 1584, 1123, 493, 171, 35, 1;

%p b:= proc(n, k, m, l) option remember; `if`(n<1, 1,

%p      `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))

%p     end:

%p A:= (n, k)-> b(n-1, min(k, n-1), 1, n):

%p T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):

%p seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);

%t b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]];

%t A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];

%t T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883.

%Y Row sums give A000110.

%Y T(2n,n) gives A322884.

%Y Cf. A287213, A287216, A287416, A287640.

%K nonn,tabf

%O 0,6

%A _Alois P. Heinz_, May 21 2017