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A136123 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).

%I A136123 #19 Jan 30 2025 05:21:31
%S A136123 1,1,1,1,3,3,11,12,1,53,56,11,309,321,87,3,2119,2175,693,53,16687,
%T A136123 17008,5934,680,11,148329,150504,55674,8064,309,1468457,1485465,
%U A136123 572650,96370,5805,53,16019531,16170035,6429470,1200070,95575,2119
%N A136123 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k maximal strings of increasing consecutive integers (0<=k<=floor(n/2)).
%C A136123 Row n has 1+floor(n/2) terms. Row sums are the factorials (A000142). Column 0 yields A000255. Column 1 yields A001277. Column 2 yields A001278. Column 3 yields A001279. Column 4 yields A001280. Sum(k*T(n,k),k>=0)=(n-2)!*(n^2 - 3n + 3)=A001564(n-2).
%D A136123 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.
%F A136123 G.f.: G(x,t) = Sum_{n>=0} n!*(((1-t)*x^2 - x)/((1-t)*x^2-1))^n. - _Vladeta Jovovic_
%e A136123 T(3,0)=3 because we have 132, 213 and 321; T(6,3)=3 because we have 125634, 341256, 563412.
%e A136123 Triangle starts:
%e A136123     1;
%e A136123     1;
%e A136123     1,   1;
%e A136123     3,   3;
%e A136123    11,  12,  1;
%e A136123    53,  56, 11;
%e A136123   309, 321, 87, 3;
%e A136123   ...
%p A136123 G:=Sum(factorial(n)*(((1-t)*x^2-x)/((1-t)*x^2-1))^n, n=0..infinity): Gser:= simplify(series(G,x=0,13)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,x,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
%p A136123 # alternative
%p A136123 A136123 := proc(n,k)
%p A136123     add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;
%p A136123     coeftayl(%,x=0,n) ;
%p A136123     coeftayl(%,y=0,k) ;
%p A136123 end proc:
%p A136123 seq(seq( A136123(n,k),k=0..floor(n/2)),n=0..12) ; # _R. J. Mathar_, Jul 01 2022
%t A136123 T[n_, k_] := Sum[x^i*(((1-y)*x-1)/((1-y)*x^2-1))^i*i!, {i, 0, n+1}] //
%t A136123    SeriesCoefficient[#, {x, 0, n}]& //
%t A136123    SeriesCoefficient[#, {y, 0, k}]&;
%t A136123 Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 09 2023, after _R. J. Mathar_ *)
%Y A136123 Cf. A000142, A000255, A001277, A001278, A001279, A001280, A001564, A010029 (rows reversed).
%K A136123 nonn,tabf
%O A136123 0,5
%A A136123 _Emeric Deutsch_ and _Vladeta Jovovic_, Dec 17 2007