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A287213 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

%I A287213 #37 Dec 27 2018 22:43:03
%S A287213 1,1,1,1,1,3,1,1,7,5,2,1,15,18,13,5,1,31,57,61,38,15,1,63,169,248,215,
%T A287213 129,52,1,127,482,935,1061,836,495,203,1,255,1341,3368,4835,4789,3573,
%U A287213 2108,877,1,511,3669,11777,20973,25430,22986,16657,9831,4140
%N A287213 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.
%C A287213 The maximal absolute difference is assumed to be zero if there is no block with consecutive elements.
%C A287213 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 A287213 Alois P. Heinz, <a href="/A287213/b287213.txt">Rows n = 0..23, flattened</a>
%H A287213 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287213 T(n,k) = A287214(n,k) - A287214(n,k-1) for k>0, T(n,0) = 1.
%e A287213 T(4,0) = 1: 1|2|3|4.
%e A287213 T(4,1) = 7: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34.
%e A287213 T(4,2) = 5: 124|3, 134|2, 13|24, 13|2|4, 1|24|3.
%e A287213 T(4,3) = 2: 14|23, 14|2|3.
%e A287213 Triangle T(n,k) begins:
%e A287213   1;
%e A287213   1;
%e A287213   1,   1;
%e A287213   1,   3,   1;
%e A287213   1,   7,   5,   2;
%e A287213   1,  15,  18,  13,    5;
%e A287213   1,  31,  57,  61,   38,  15;
%e A287213   1,  63, 169, 248,  215, 129,  52;
%e A287213   1, 127, 482, 935, 1061, 836, 495, 203;
%p A287213 b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
%p A287213       `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
%p A287213       `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
%p A287213     end:
%p A287213 A:= (n, k)-> b(n, min(k, n-1), []):
%p A287213 T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
%p A287213 seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);
%t A287213 b[0, _, _] = 1; b[n_, k_, l_] := b[n, k, l] =b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}];
%t A287213 A[n_, k_] := b[n, Min[k, n - 1], {}];
%t A287213 T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A287213 Table[Table[T[n, k], {k, 0, Max[n - 1, 0]}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A287213 Columns k=0-10 give: A000012, A000225(n-1), A258109, A294052, A294053, A294054, A294055, A294056, A294057, A294058, A294059.
%Y A287213 Row sums and T(n+2,n+1) give A000110.
%Y A287213 T(2n,n) gives A294024.
%Y A287213 Cf. A287214, A287215, A287416, A287640.
%K A287213 nonn,tabf
%O A287213 0,6
%A A287213 _Alois P. Heinz_, May 21 2017