A134362 -id:A134362 - cp's OEIS frontend

A208231 Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

0, 1, 5, 37, 373, 4761, 73601, 1336609, 27888281, 657386305, 17276807089, 500876786301, 15879053677697, 546470462226313, 20288935994319929, 808320431258439121, 34397370632215764001, 1557106493482564625793, 74713970491718324746529, 3787792171563440619543133, 202314171910557294992453009

Offset: 0

Geoffrey Critzer, Jan 10 2013

nonn

Sum of the number of endofunctions whose cycle lengths are >=i for all i >=1. A000312 + A065440 + A134362 + A208230 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000169, A028417, A000312, A065440, A134362, A208230, A208248, A290932.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j$2)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 20 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Apply[Plus,Table[Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]]

E.g.f.: A(T(x)) = Sum_{k>=1} exp( Sum_{i>=k} T(x)^i/i) - 1 where A(x) is the e.g.f. for A028417 and T(x) is the e.g.f. for A000169.

A208230 Number of functions f:{1,2,...,n}->{1,2,...,n} with all cycles of length >= 4.

Original entry on oeis.org

1, 0, 0, 0, 6, 144, 3000, 64560, 1498140, 37906848, 1046608416, 31438821600, 1023129229320, 35910464987760, 1353422643322464, 54548490915316944, 2342204085734058000, 106771822456475695680, 5151207243866077428480, 262261296920723111462592, 14053817061169685865626976

Offset: 0

Geoffrey Critzer, Jan 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000312, A065440, A134362.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1)*(j-1)!, j=4..n))
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, May 20 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,4,nn}]],{x,0,nn}],x]

E.g.f.: exp( Sum_{i>=4} T(x)^i/i ) where T(x) is the e.g.f. for A000169

A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 2, 1, 3, 8, 18, 30, 1, 4, 15, 52, 163, 444, 1, 5, 24, 110, 478, 1950, 7360, 1, 6, 35, 198, 1083, 5706, 28821, 138690, 1, 7, 48, 322, 2110, 13482, 83824, 505876, 2954364, 1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569, 70469000, 1, 9, 80, 702, 6078, 51894, 436656, 3618540, 29510268, 236644092, 1864204416, 1, 10, 99, 970, 9403, 90150, 854485, 8003950, 74058105, 676549450, 6098971555, 54224221050

Offset: 0

Mason C. Hart, Apr 14 2020

Comes up in the study of the Zen Stare game (see description at A134362).

T(k,n-k)*binomial(n,k)*(n-k-1)!! is the number of different possible Zen Stare rounds with n starting players and k winners.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     1     1      1      1      1       1 ...
  1 |    0     1     2      3      4      5       6 ...
  2 |    0     3     8     15     24     35      48 ...
  3 |    2    18    52    110    198    322     488 ...
  4 |   30   163   478   1083   2110   3715    6078 ...
  5 |  444  1950  5706  13482  27768  51894   90150 ...
  6 | 7360 28821 83824 203569 436656 854485 1557376 ...
  ...
T(2,2) = 8; This because given X = {A,B}, Y = {A,B,C,D}. The only functions f: X->Y that meet the requirement are:
f(A) = C, f(B) = C
f(A) = D, f(B) = D
f(A) = D, f(B) = C
f(A) = C, f(B) = D
f(A) = B, f(B) = C
f(A) = B, f(B) = D
f(A) = C, f(B) = A
f(A) = D, f(B) = A

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..3 are A000012, A001477, A005563, A058794.
Columns k=0..4 are A134362, A089466, A089467, A089468, A220690(n+2).
Cf. A000169, A065440.

T(n,k)={my(w=-lambertw(-x + O(x^max(4,1+n)))); n!*polcoef(exp((k-1)*w - w^2/2)/(1-w), n)} \\ Andrew Howroyd, Apr 15 2020

T(n,k) = Sum_{i=0..n} k^(n-i)*binomial(n,i)*T(i,n-i); This means that with a constant n, T(n,k) is a polynomial of k.
T(n,0) = A134362(n).
T(0,k) = 1.
For odd n, Sum_{k=1..(n+1)/2} T(2*k-1,n-2*k+1)*binomial(n,2*k-1)*(n-2*k)!! = (n-1)^n.
E.g.f. of k-th column: exp((k-1)*W(x) - W(x)^2/2)/(1-W(x)) where W(x) is the e.g.f. of A000169. - Andrew Howroyd, Apr 15 2020

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A208231 Sum of the minimum cycle length over all functions f:{1,2,...,n}->{1,2,...,n} (endofunctions).

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A208230 Number of functions f:{1,2,...,n}->{1,2,...,n} with all cycles of length >= 4.

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A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.

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