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A320554 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most four elements and for at least one block c the smallest integer interval containing c has exactly four elements.

%I A320554 #12 Dec 08 2020 08:37:11
%S A320554 5,17,45,121,336,901,2347,6014,15314,38766,97531,244054,608339,
%T A320554 1511919,3748379,9273353,22901665,56477538,139114445,342325451,
%U A320554 841676972,2067997764,5078117000,12463618356,30577931115,74993361731,183870516407,450708620604,1104563863868
%N A320554 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most four elements and for at least one block c the smallest integer interval containing c has exactly four elements.
%H A320554 Alois P. Heinz, <a href="/A320554/b320554.txt">Table of n, a(n) for n = 4..1000</a>
%F A320554 G.f.: -(-x^8 -3*x^7 +x^6 +7*x^5 +5*x^4) / (x^12 +5*x^11 +7*x^10 -x^9 -13*x^8 -19*x^7 -14*x^6 -6*x^5 +x^4 +2*x^2 +2*x-1).
%F A320554 a(n) = A276720(n) - A129847(n).
%e A320554 a(5) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.
%p A320554 b:= proc(n, m, l) option remember; `if`(n=0, 1,
%p A320554       add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
%p A320554       `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
%p A320554     end:
%p A320554 A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
%p A320554 a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(4):
%p A320554 seq(a(n), n=4..35);
%t A320554 b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];
%t A320554 A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
%t A320554 a[n_] := With[{k = 4}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
%t A320554 a /@ Range[4, 35] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A320554 Column k=4 of A276727.
%Y A320554 Cf. A129847, A276720, A320617.
%K A320554 nonn,easy
%O A320554 4,1
%A A320554 _Alois P. Heinz_, Oct 15 2018