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.
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
Examples
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
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325
Crossrefs
Programs
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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
Formula
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
Comments