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A350275
Irregular triangle read by rows: T(n,k) is the number of endofunctions on [n] whose fourth-largest component has size exactly k; n >= 0, 0 <= k <= floor(n/4).
6
1, 1, 4, 27, 255, 1, 3094, 31, 45865, 791, 803424, 20119, 16239720, 528991, 8505, 372076163, 14689441, 654885, 9529560676, 435580164, 34859160, 269819334245, 13846282341, 1646054025, 8369112382488, 471890017358, 73811825010, 1286223400
OFFSET
0,3
COMMENTS
An endofunction on [n] is a function from {1,2,...,n} to {1,2,...,n}.
If the mapping has no fourth component, then its fourth-largest component is defined to have size 0.
LINKS
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.
EXAMPLE
Triangle begins:
1;
1;
4;
27;
255, 1;
3094, 31;
45865, 791;
803424, 20119;
...
MAPLE
g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add(g(i)*
b(n-i, sort([l[], i])[-4..-1])*binomial(n-1, i-1), i=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14); # Alois P. Heinz, Dec 22 2021
MATHEMATICA
g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[g[i]*b[n - i, Sort[ Append[l, i]][[-4 ;; -1]]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)
CROSSREFS
Row sums give A000312.
Sequence in context: A243696 A158836 A350276 * A239380 A239379 A239298
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Dec 22 2021
STATUS
approved