A161127 - OEIS

%I #20 Sep 05 2024 12:21:14

%S 1,1,1,1,1,3,7,3,1,1,6,27,37,27,6,1,1,10,76,220,331,220,76,10,1,1,15,

%T 176,897,2438,3341,2438,897,176,15,1,1,21,357,2885,12825,30807,41343,

%U 30807,12825,2885,357,21,1,1,28,658,7871,53312,203927,452931,589569

%N Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k descents (n>=1; 1<=k<2n-1).

%C Row n contains 2n-1 entries.

%C Sum of entries in row n = (2n-1)!! = A001147(n).

%C Sum_{k=1..2n-1} k*T(n,k) = A001879(n-1).

%H J. Désarménien and D. Foata, <a href="http://www.numdam.org/item?id=BSMF_1985__113__3_0">Fonctions symetriques et series hypergeometriques basiques multivariees</a>, Bull. Soc. Math. France, 113, 1985, 3-22.

%H I. M. Gessel and C. Reutenauer, <a href="http://dx.doi.org/10.1016/0097-3165(93)90095-P">Counting permutations with given cycle structure and descent set</a>, J. Combin. Theory, Ser. A, 64, 1993, 189-215.

%H V. J. W. Guo and J. Zeng, <a href="http://dx.doi.org/10.1016/j.jcta.2005.10.002">The Eulerian distribution on involutions is indeed unimodal</a>, J. Combin. Theory, Ser. A, 113, 2006, 1061-1071.

%F Recurrence: 2nT(n,k) = [k(k+1)+2n-2]T(n-1,k)+2[(k-1)(2n-k-1)+1]T(n-1,k-1)+[(2n-k)(2n-k+1)+2n-2]T(n-1,k-2) (k>=1, n>=2) (see Eq. (2.1) in the Guo-Zeng paper; see first Maple program).

%F Generating polynomial of row n is P(n,t) = (1 - t)^{2n+1}*Sum(C(j(j+1)/2 + n - 1, n)*t^j, j=0..oo) (see Eq. (2.2) in the Guo-Zeng paper; see 2nd Maple program).

%e T(3,2)=3 because we have 215634, 341265, and 351624.

%e Triangle starts:

%e   1;

%e   1,1,1;

%e   1,3,7,3,1;

%e   1,6,27,37,27,6,1;

%e   1,10,76,220,331,220,76,10,1;

%e   ...

%p T := proc (n, k) if k <= 0 or n <= 0 then 0 elif n = 1 and k = 1 then 1 elif 2*n <= k then 0 else ((1/2)*(k*(k+1)+2*n-2)*T(n-1, k)+(1/2)*(2*(k-1)*(2*n-k-1)+2)*T(n-1, k-1)+(1/2)*((2*n-k)*(2*n-k+1)+2*n-2)*T(n-1, k-2))/n end if end proc: for n to 8 do seq(T(n, k), k = 1 .. 2*n-1) end do; # end of program

%p for n to 8 do P[n] := sort(expand(simplify((1-t)^(2*n+1)*(sum(binomial((1/2)*i*(i+1)+n-1, n)*t^i, i = 0 .. infinity))))) end do: for n to 8 do seq(coeff(P[n], t, j), j = 1 .. 2*n-1) end do; # end of program

%t T[n_, k_] := Which[k <= 0 || n <= 0, 0, n == 1 && k == 1, 1, 2n <= k, 0, True, ((1/2)*(k*(k + 1) + 2n - 2)*T[n - 1, k] + (1/2)*(2*(k - 1)*(2n - k - 1) + 2)*T[n - 1, k - 1] + (1/2)*((2n - k)*(2n - k + 1) + 2n - 2)*T[n - 1, k - 2])/n];

%t Table[T[n, k], {n, 1, 8}, {k, 1, 2n - 1}] // Flatten (* _Jean-François Alcover_, Sep 05 2024, after first Maple program *)

%Y Cf. A001147, A001879.

%K nonn,tabf

%O 1,6

%A _Emeric Deutsch_, Jun 09 2009