%I #77 Sep 18 2021 10:51:32

%S 0,1,1,1,1,7,1,1,21,21,1,1,51,161,51,1,1,113,813,813,113,1,1,239,3361,

%T 7631,3361,239,1,1,493,12421,53833,53833,12421,493,1,1,1003,42865,

%U 320107,607009,320107,42865,1003,1,1,2025,141549,1704693,5494017

%N Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

%C From _Emeric Deutsch_, May 25 2009: (Start)

%C T(n,k) is the number of derangements of [n] having k excedances. Example: T(4,2)=7 because we have 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 3*4*21, 4*3*21, each with two excedances (marked). An excedance of a permutation p is a position i such that p(i) > i.

%C Sum_{k>=1} k*T(n,k) = A000274(n+1).

%C (End)

%C The triangle 1;1,1;1,7,1;... has general term T(n,k) = Sum_{j=0..n+2} (-1)^(n-j)*C(n+2,j)*A123125(j,k+2) and bivariate g.f. ((1-y)*(y*exp(2*x*y) + exp(x*(y+1))(y^2 - 4*y + 1) + y*exp(2*x)))/(exp(x*y) - y*exp(x))^3. - _Paul Barry_, May 10 2011

%C The n-th row is the local h-vector of the barycentric subdivision of a simplex, i.e., the Coxeter complex of type A. See Proposition 2.4 of Stanley's paper below. - _Kyle Petersen_, Aug 20 2012

%C T(n,k) is the k-th coefficient of the local h^*-polynomial, or box polynomial, of the s-lecture hall n-simplex with s=(2,3,...,n+1). See Theorem 4.1 of the paper by N. Gustafsson and L. Solus below. - _Liam Solus_, Aug 23 2018

%H Robert Israel, <a href="/A046739/b046739.txt">Table of n, a(n) for n = 1..10012</a> (rows 0 to 142, flattened)

%H L. Carlitz, 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 L. Carlitz, N. J. A. Sloane, and C. L. Mallows, <a href="/A259465/a259465_1.pdf">Correspondence, 1975</a>

%H N. Gustafsson and L. Solus. <a href="https://arxiv.org/abs/1807.05246">Derangements, Ehrhart theory, and local h-polynomials</a>, arXiv:1807.05246 [math.CO], 2018.

%H R. Mantaci and F. Rakotondrajao, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00531-6">Exceedingly deranging!</a>, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]

%H D. P. Roselle, <a href="/A046739/a046739.pdf"> Permutations by number of rises and successions</a>, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]

%H D. P. Roselle, <a href="http://www.jstor.org/stable/2036129">Permutations by number of rises and successions</a>, Proc. Amer. Math. Soc., 20 (1968), 8-16.

%H R. P. Stanley, <a href="http://dx.doi.org/10.1090/S0894-0347-1992-1157293-9">Subdivisions and local h-vectors</a>, J. Amer. Math. Soc., 5 (1992), 805-851.

%H John D. Wiltshire-Gordon, Alexander Woo, and Magdalena Zajaczkowska, <a href="https://arxiv.org/abs/1701.05277">Specht Polytopes and Specht Matroids</a>, arXiv:1701.05277 [math.CO], 2017. [See Conjecture 6.2]

%F a(n+1, r) = r*a(n, r) + (n+1-r)*a(n, r-1) + n*a(n-1, r-1).

%F exp(-t)/(1 - exp((x-1)t)/(x-1)) = 1 + x*t^2/2! + (x+x^2)*t^3/3! + (x+7x^2+x^3)*t^4/4! + (x+21x^2+21x^3+x^4)*t^5/5! + ... - _Philippe Deléham_, Jun 11 2004

%e Triangle starts:

%e 0;

%e 1;

%e 1, 1;

%e 1, 7, 1;

%e 1, 21, 21, 1;

%e 1, 51, 161, 51, 1;

%e 1, 113, 813, 813, 113, 1;

%e ...

%e From _Peter Luschny_, Sep 17 2021: (Start)

%e The triangle shows the coefficients of the following bivariate polynomials:

%e [1] 0;

%e [2] x*y;

%e [3] x^2*y + x*y^2;

%e [4] x^3*y + 7*x^2*y^2 + x*y^3;

%e [5] x^4*y + 21*x^3*y^2 + 21*x^2*y^3 + x*y^4;

%e [6] x^5*y + 51*x^4*y^2 + 161*x^3*y^3 + 51*x^2*y^4 + x*y^5;

%e [7] x^6*y + 113*x^5*y^2 + 813*x^4*y^3 + 813*x^3*y^4 + 113*x^2*y^5 + x*y^6;

%e ...

%e These polynomials are the permanents of the n X n matrices with all entries above the main antidiagonal set to 'x' and all entries below the main antidiagonal set to 'y'. The main antidiagonals consist only of zeros. Substituting x <- 1 and y <- -1 generates the Euler secant numbers A122045. (Compare with A081658.)

%e (End)

%p G := (1-t)*exp(-t*z)/(1-t*exp((1-t)*z)): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form # _Emeric Deutsch_, May 25 2009

%t max = 12; f[t_, z_] := (1-t)*(Exp[-t*z]/(1 - t*Exp[(1-t)*z])); se = Series[f[t, z], {t, 0, max}, {z, 0, max}];

%t coes = Transpose[ #*Range[0, max]! & /@ CoefficientList[se, {t, z}]]; Join[{0}, Flatten[ Table[ coes[[n, k]], {n, 2, max}, {k, 2, n-1}]]] (* _Jean-François Alcover_, Oct 24 2011, after g.f. *)

%t E1[n_ /; n >= 0, 0] = 1; (* E1(n,k) are the Eulerian numbers *)

%t E1[n_, k_] /; k < 0 || k > n = 0;

%t E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];

%t T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];

%t Table[T[n, k], {n, 1, 100}, {k, 1, n-1}] /. {} -> {0} // Flatten (* _Jean-François Alcover_, Oct 31 2020, after _Peter Luschny_ in A271697 *)

%Y Cf. A046740. Row sums give A000166. Diagonals give A070313, A070315.

%Y Cf. A000274, A081658, A122045, A271697.

%K nonn,easy,nice,tabf

%O 1,6

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000