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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.
3
5, 17, 45, 121, 336, 901, 2347, 6014, 15314, 38766, 97531, 244054, 608339, 1511919, 3748379, 9273353, 22901665, 56477538, 139114445, 342325451, 841676972, 2067997764, 5078117000, 12463618356, 30577931115, 74993361731, 183870516407, 450708620604, 1104563863868
OFFSET
4,1
LINKS
FORMULA
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).
a(n) = A276720(n) - A129847(n).
EXAMPLE
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.
MAPLE
b:= proc(n, m, l) option remember; `if`(n=0, 1,
add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(4):
seq(a(n), n=4..35);
MATHEMATICA
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}}]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
a[n_] := With[{k = 4}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
a /@ Range[4, 35] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)
CROSSREFS
Column k=4 of A276727.
Sequence in context: A247618 A269962 A048612 * A218135 A271122 A147050
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Oct 15 2018
STATUS
approved