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A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.

%I A088310 #21 Dec 15 2022 05:18:45
%S A088310 1,2,10,264,33864,19158720,44680224960,413586858182400,
%T A088310 14960200449325582080,2109063823453947981680640,
%U A088310 1162864344149083760773678387200,2520991223487759548686737154649702400,21598422878151131130336454273775859841843200,734233037731110118818452425552296701963294284185600
%N A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.
%H A088310 G. C. Greubel, <a href="/A088310/b088310.txt">Table of n, a(n) for n = 0..57</a>
%F A088310 a(n) = n! * Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - _Vladeta Jovovic_, Nov 07 2003
%F A088310 a(n) = Sum_{i=0..n} Sum_{j=0..n} stirling1(n, i) * stirling1(n, j) * 2^(i*j). - _Max Alekseyev_, Nov 07 2003
%F A088310 a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Jul 02 2016
%F A088310 a(n) = A181230(n,n).
%e A088310 a(2) = 10: 00/01, 00/10, 01/00, 01/10, 01/11, 10/00, 10/01, 10/11, 11/01, 11/10.
%t A088310 Table[n!*Sum[StirlingS1[n, k]*Binomial[2^k,n], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)
%o A088310 (Magma)
%o A088310 A088310:= func< n | Factorial(n)*(&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
%o A088310 [A088310(n): n in [0..30]]; // _G. C. Greubel_, Dec 14 2022
%o A088310 (SageMath)
%o A088310 @CachedFunction
%o A088310 def A088310(n): return (-1)^n*factorial(n)*sum((-1)^k*binomial(2^k,n)*stirling_number1(n,k) for k in (0..n))
%o A088310 [A088310(n) for n in range(31)] # _G. C. Greubel_, Dec 14 2022
%Y A088310 Cf. A088229, A088309.
%Y A088310 Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
%K A088310 nonn
%O A088310 0,2
%A A088310 _N. J. A. Sloane_, Nov 07 2003
%E A088310 Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003
%E A088310 a(0)-a(5) from _W. Edwin Clark_, Nov 07 2003