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A362763 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-sets of permutations of an n-set.

%I A362763 #15 Apr 09 2025 11:22:09
%S A362763 1,1,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,0,5,5,1,0,0,0,6,23,7,1,0,0,0,5,116,
%T A362763 89,11,1,0,0,0,3,521,2494,484,15,1,0,0,0,1,1931,69366,87984,2904,22,1,
%U A362763 0,0,0,0,5906,1592714,15456557,4250015,22002,30,1
%N A362763 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-sets of permutations of an n-set.
%C A362763 Isomorphism is up to permutation of the elements of the n-set. Each permutation can be considered to be a set of disjoint directed cycles whose vertices cover the n-set. Permuting the elements of the n-set permutes each of the permutations in the k-set.
%H A362763 Andrew Howroyd, <a href="/A362763/b362763.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A362763 Aman Kushwaha and Raghavendra Tripathi, <a href="https://arxiv.org/abs/2503.09542">A note on Erdős matrices and Marcus-Ree inequality</a>, arXiv:2503.09542 [math.MG], 2025. See p. 12.
%F A362763 T(n,k) = 0 for k > n!.
%F A362763 T(n,k) = T(n, n!-k).
%e A362763 Array begins:
%e A362763 ====================================================================
%e A362763 n/k| 0  1    2       3          4             5                6 ...
%e A362763 ---+----------------------------------------------------------------
%e A362763 0  | 1  1    0       0          0             0                0 ...
%e A362763 1  | 1  1    0       0          0             0                0 ...
%e A362763 2  | 1  2    1       0          0             0                0 ...
%e A362763 3  | 1  3    5       6          5             3                1 ...
%e A362763 4  | 1  5   23     116        521          1931             5906 ...
%e A362763 5  | 1  7   89    2494      69366       1592714         30461471 ...
%e A362763 6  | 1 11  484   87984   15456557    2209040882     263190866673 ...
%e A362763 7  | 1 15 2904 4250015 5329123475 5366409944453 4503264576070573 ...
%e A362763   ...
%o A362763 (PARI)
%o A362763 B(n,k) = {n!*k^n}
%o A362763 K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
%o A362763 R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
%o A362763 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A362763 T(n,k) = {if(n==0, k<=1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(R(q,m))*(x^m-x^(2*m))/m, O(x*x^k))), k)); s/n!)}
%Y A362763 Columns k=0..3 are A000012, A000041, A362764, A362765.
%Y A362763 Row sums are A362766.
%Y A362763 Cf. A362644.
%K A362763 nonn,tabl
%O A362763 0,9
%A A362763 _Andrew Howroyd_, May 03 2023