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A259465 Triangle read by rows: enumerates pairs of amicable permutations by rises.

%I A259465 #38 Jul 09 2025 04:39:55
%S A259465 1,1,1,1,1,8,1,1,43,43,1,1,194,826,194,1,1,803,11284,11284,803,1,1,
%T A259465 3184,127905,392244,127905,3184,1,1,12367,1297629,10258067,10258067,
%U A259465 1297629,12367,1,1,47606,12295720,224702858,561134638,224702858,12295720,47606,1
%N A259465 Triangle read by rows: enumerates pairs of amicable permutations by rises.
%H A259465 Alois P. Heinz, <a href="/A259465/b259465.txt">Rows n = 0..80, flattened</a>
%H A259465 Carlitz, L., Richard Scoville, and Theresa Vaughan, <a href="/A259465/a259465.pdf">Enumeration of pairs of permutations and sequences</a>,  Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
%H A259465 L. Carlitz, N. J. A. Sloane, and C. L. Mallows, <a href="/A259465/a259465_1.pdf">Correspondence, 1975</a>
%e A259465 Triangle begins:
%e A259465   1;
%e A259465   1;
%e A259465   1,     1;
%e A259465   1,     8,       1;
%e A259465   1,    43,      43,        1;
%e A259465   1,   194,     826,      194,        1;
%e A259465   1,   803,   11284,    11284,      803,       1;
%e A259465   1,  3184,  127905,   392244,   127905,    3184,     1;
%e A259465   1, 12367, 1297629, 10258067, 10258067, 1297629, 12367, 1;
%e A259465   ...
%p A259465 b:= proc(u, o, h) option remember; `if`(u+o=0, 1, expand(
%p A259465       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h)*x, j=1..u)+
%p A259465       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o)))
%p A259465     end:
%p A259465 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n$2)):
%p A259465 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jul 02 2015
%t A259465 b[u_, o_, h_] := b[u, o, h] = If[u+o == 0, 1, Expand[Sum[Sum[b[u-j, o+j-1, h+i-1], {i, 1, u+o-h}]*x, {j, 1, u}] + Sum[Sum[b[u+j-1, o-j, h-i], {i, 1, h}], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[0, n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 12 2016, after _Alois P. Heinz_ *)
%Y A259465 Row sums give A060350.
%K A259465 nonn,tabf
%O A259465 0,6
%A A259465 _N. J. A. Sloane_, Jun 30 2015, following a suggestion from L. Carlitz, Nov 30 1975
%E A259465 More terms from _Alois P. Heinz_, Jul 02 2015