A334014 - OEIS

%I #38 May 11 2020 02:19:29

%S 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,

%T 1950,7360,1,6,35,198,1083,5706,28821,138690,1,7,48,322,2110,13482,

%U 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

%N 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.

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

%C 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.

%H Andrew Howroyd, <a href="/A334014/b334014.txt">Table of n, a(n) for n = 0..1325</a>

%F 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.

%F T(n,0) = A134362(n).

%F T(0,k) = 1.

%F 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.

%F 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

%e Array begins:

%e =======================================================

%e n\k |    0     1     2      3      4      5       6

%e ----+--------------------------------------------------

%e   0 |    1     1     1      1      1      1       1 ...

%e   1 |    0     1     2      3      4      5       6 ...

%e   2 |    0     3     8     15     24     35      48 ...

%e   3 |    2    18    52    110    198    322     488 ...

%e   4 |   30   163   478   1083   2110   3715    6078 ...

%e   5 |  444  1950  5706  13482  27768  51894   90150 ...

%e   6 | 7360 28821 83824 203569 436656 854485 1557376 ...

%e   ...

%e 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:

%e f(A) = C, f(B) = C

%e f(A) = D, f(B) = D

%e f(A) = D, f(B) = C

%e f(A) = C, f(B) = D

%e f(A) = B, f(B) = C

%e f(A) = B, f(B) = D

%e f(A) = C, f(B) = A

%e f(A) = D, f(B) = A

%o (PARI) 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

%Y Rows n=0..3 are A000012, A001477, A005563, A058794.

%Y Columns k=0..4 are A134362, A089466, A089467, A089468, A220690(n+2).

%Y Cf. A000169, A065440.

%K nonn,tabl

%O 0,8

%A _Mason C. Hart_, Apr 14 2020