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A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A287214 #35 Dec 27 2018 20:43:17
%S A287214 1,1,1,1,1,1,1,1,2,1,1,1,2,4,1,1,1,2,5,8,1,1,1,2,5,13,16,1,1,1,2,5,15,
%T A287214 34,32,1,1,1,2,5,15,47,89,64,1,1,1,2,5,15,52,150,233,128,1,1,1,2,5,15,
%U A287214 52,188,481,610,256,1,1,1,2,5,15,52,203,696,1545,1597,512,1
%N A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A287214 The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.
%H A287214 Alois P. Heinz, <a href="/A287214/b287214.txt">Antidiagonals n = 0..45, flattened</a>
%H A287214 Pierpaolo Natalini, Paolo Emilio Ricci, <a href="https://doi.org/10.3390/axioms7040071">New Bell-Sheffer Polynomial Sets</a>, Axioms 2018, 7(4), 71.
%H A287214 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287214 A(n,k) = Sum_{j=0..k} A287213(n,j).
%e A287214 A(4,0) = 1: 1|2|3|4.
%e A287214 A(4,1) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
%e A287214 A(4,2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
%e A287214 Square array A(n,k) begins:
%e A287214   1,  1,   1,   1,   1,   1,   1,   1, ...
%e A287214   1,  1,   1,   1,   1,   1,   1,   1, ...
%e A287214   1,  2,   2,   2,   2,   2,   2,   2, ...
%e A287214   1,  4,   5,   5,   5,   5,   5,   5, ...
%e A287214   1,  8,  13,  15,  15,  15,  15,  15, ...
%e A287214   1, 16,  34,  47,  52,  52,  52,  52, ...
%e A287214   1, 32,  89, 150, 188, 203, 203, 203, ...
%e A287214   1, 64, 233, 481, 696, 825, 877, 877, ...
%p A287214 b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
%p A287214       `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
%p A287214       `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
%p A287214     end:
%p A287214 A:= (n, k)-> b(n, min(k, n-1), []):
%p A287214 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A287214 b[0, _, _] = 1; b[n_, k_, l_List] := 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 A287214 A[n_, k_] := b[n, Min[k, n - 1], {}];
%t A287214 Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A287214 Columns k=0-10 give: A000012, A011782, A001519, A287275, A287276, A287277, A287278, A287279, A287280, A287281, A287282.
%Y A287214 Main diagonal gives A000110.
%Y A287214 Cf. A287213, A287216, A287417, A287641.
%K A287214 nonn,tabl
%O A287214 0,9
%A A287214 _Alois P. Heinz_, May 21 2017