A320337 -id:A320337 - cp's OEIS frontend

A046739 Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

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, 7631, 3361, 239, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 1, 2025, 141549, 1704693, 5494017

Offset: 1

N. J. A. Sloane

From Emeric Deutsch, May 25 2009: (Start)
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.
Sum_{k>=1} k*T(n,k) = A000274(n+1). (End)
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
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
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

			Triangle starts:
  0;
  1;
  1,   1;
  1,   7,   1;
  1,  21,  21,   1;
  1,  51, 161,  51,   1;
  1, 113, 813, 813, 113, 1;
  ...
From _Peter Luschny_, Sep 17 2021: (Start)
The triangle shows the coefficients of the following bivariate polynomials:
  [1] 0;
  [2] x*y;
  [3] x^2*y +     x*y^2;
  [4] x^3*y +   7*x^2*y^2 +     x*y^3;
  [5] x^4*y +  21*x^3*y^2 +  21*x^2*y^3 +     x*y^4;
  [6] x^5*y +  51*x^4*y^2 + 161*x^3*y^3 +  51*x^2*y^4 +     x*y^5;
  [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;
  ...
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.)
(End)

Robert Israel, Table of n, a(n) for n = 1..10012 (rows 0 to 142, flattened)
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975
N. Gustafsson and L. Solus. Derangements, Ehrhart theory, and local h-polynomials, arXiv:1807.05246 [math.CO], 2018.
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]
Lili Mu and Volkmar Welker, On a question about real rooted polynomials and f-polynomials of simplicial complexes, arXiv:2503.24076 [math.CO], 2025. See p. 8.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.
R. P. Stanley, Subdivisions and local h-vectors, J. Amer. Math. Soc., 5 (1992), 805-851.
John D. Wiltshire-Gordon, Alexander Woo, and Magdalena Zajaczkowska, Specht Polytopes and Specht Matroids, arXiv:1701.05277 [math.CO], 2017. [See Conjecture 6.2]

Cf. A046740.
Row sums give A000166.
Diagonals give A070313, A070315.
T(2n,n) gives A320337.
Cf. A000274, A081658, A122045, A271697.

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

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}];
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. *)
E1[n_ /; n >= 0, 0] = 1; (* E1(n,k) are the Eulerian numbers *)
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
Table[T[n, k], {n, 1, 100}, {k, 1, n-1}] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 31 2020, after Peter Luschny in A271697 *)
Table[Expand[n!Factor[SeriesCoefficient[(x-y)/(x Exp[y t]-y Exp[x t]),{t,0,n}]]],{n,0,12}]//TableForm (* Mamuka Jibladze, Nov 26 2024 *)

T(n)={my(x='x+O('x^(n+1))); concat([[0]], [Vecrev(p/y) | p<-Vec(-1+serlaplace((y-1)/(y*exp(x)-exp(x*y))))])}
{ my(A=T(10));for(i=1,#A,print(A[i])) } \\ Andrew Howroyd, Nov 13 2024

a(n+1, r) = r*a(n, r) + (n+1-r)*a(n, r-1) + n*a(n-1, r-1).
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.g.f.: (y-1)/(y*exp(x) - exp(x*y)). - Mamuka Jibladze, Nov 08 2024

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

A062868 Number of permutations of degree n with barycenter 0.

Original entry on oeis.org

1, 1, 2, 4, 14, 46, 282, 1394, 12658, 83122, 985730, 8012962, 116597538, 1127575970, 19410377378, 217492266658, 4320408974978, 55023200887938, 1238467679662722, 17665859065690754, 444247724347355554, 7015393325151055906, 194912434760367113570, 3375509056735963889634

Offset: 0

Olivier Gérard, Jun 26 2001

The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.

			(4,1,3,5,2) has difference (3,-1,0,1,-3) and signs (1,-1,0,1,-1) with total 0.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 151 terms from Maxwell Jiang)

Column k=0 of A062866 or of A062867.

Cf. A179567, A196687, A196688, A320337.

b:= proc(s, t) option remember; (n-> `if`(abs(t)>n, 0, `if`(n=0, 1,
      add(b(s minus {j}, t+signum(n-j)), j=s))))(nops(s))
    end:
a:= n-> b({$1..n}, 0):
seq(a(n), n=0..14);  # Alois P. Heinz, Jul 31 2018

E1[n_ /; n >= 0, 0] = 1;
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
b[n_] := Sum[(-1)^(n-k) E1[n+k, n] Binomial[2n, n-k], {k, 0, n}];
a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
a /@ Range[0, 150] (* Jean-François Alcover, Oct 29 2020, after Peter Luschny in A320337 *)

a(n) = Sum_{k=0..floor(n/2)} binomial(n, n-2*k)*A320337(k). - Maxwell Jiang, Dec 19 2018 (added by editors)

a(n) ~ sqrt(3) * (1 + exp(-2)*(-1)^n) * n^n / exp(n). - Vaclav Kotesovec, Oct 29 2020

One more term from Vladeta Jovovic, Jun 28 2001
a(11)-a(14) from Hugo Pfoertner, Sep 23 2004
a(15)-a(18) from R. H. Hardin, Jul 18 2010
a(19)-a(22) from Kyle G Hess, Jul 30 2018
a(0)=1 prepended by Alois P. Heinz, Jul 30 2018
Terms a(23) and beyond from Maxwell Jiang, Dec 19 2018

A271697 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 7, 1, 0, 0, 1, 21, 21, 1, 0, 0, 1, 51, 161, 51, 1, 0, 0, 1, 113, 813, 813, 113, 1, 0, 0, 1, 239, 3361, 7631, 3361, 239, 1, 0, 0, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 0, 0, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 0

Offset: 0

Peter Luschny, Apr 12 2016

			Triangle starts:
  1;
  0, 0;
  0, 1,   0;
  0, 1,   1,   0;
  0, 1,   7,   1,   0;
  0, 1,  21,  21,   1,   0;
  0, 1,  51, 161,  51,   1, 0;
  0, 1, 113, 813, 813, 113, 1, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Peter Luschny, Extensions of the binomial

Variant: A046739 (main entry for this triangle).
Cf. A000166 (row sums), A122045 (Euler numbers are the alternating row sums), A070313 (col. 2) and (diag. n,n-2).
Cf. A173018.
T(2n,n) gives A320337.

A271697 := (n,k) -> add(binomial(-j-1,-n-1)*combinat:-eulerian1(j,k), j=0..n):
seq(seq(A271697(n, k), k=0..n), n=0..11);

<= 0, 0] = 1;
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 29 2020 *)

T(n)={my(x='x+O('x^(n+1)), v=Vec(serlaplace((y-1)/(y*exp(x)-exp(x*y))))); vector(#v,n,Vecrev(v[n],n))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Nov 13 2024

T(n,k) = T(n,n-k). - Alois P. Heinz, Oct 29 2020

A321967 Triangle read by rows, T(n,k) = binomial(-k-n-1, -2n-1)E1(k+n, n), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -4, 11, 0, 15, -156, 302, 0, -56, 1596, -9528, 15619, 0, 210, -14400, 193185, -882340, 1310354, 0, -792, 122265, -3213760, 30042672, -116857368, 162512286, 0, 3003, -1005004, 47887840, -802069632, 6034981134, -21078701112, 27971176092

Offset: 0

Peter Luschny, Dec 18 2018

			Triangle starts:
                       1;
                  0,        1;
              0,      -4,       11;
          0,     15,      -156,      302;
       0,   -56,     1596,    -9528,     15619;
    0,   210,  -14400,   193185,   -882340,   1310354;
  0, -792, 122265, -3213760, 30042672, -116857368, 162512286;

Row sums give A320337.

Cf. A046739, A180056 (main diagonal), A271697, A001791.

T := (n, k) -> binomial(-k-n-1, -2*n-1)*combinat:-eulerian1(k+n, n):
for n from 0 to 7 do seq(T(n,k), k=0..n) od;

E1[n_ /; n >= 0, 0] = 1; E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n - k) E1[n - 1, k - 1] + (k + 1) E1[n - 1, k];
T[n_, k_] := Binomial[-k - n - 1, -2 n - 1] E1[n + k, n];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Dec 30 2018 *)

cp's OEIS Frontend

A046739 Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

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A062868 Number of permutations of degree n with barycenter 0.

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A271697 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

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A321967 Triangle read by rows, T(n,k) = binomial(-k-n-1, -2*n-1)*E1(k+n, n), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.

Views

Author

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Maple

Mathematica

A321967 Triangle read by rows, T(n,k) = binomial(-k-n-1, -2n-1)E1(k+n, n), E1 the Eulerian numbers A173018, for n >= 0 and 0 <= k <= n.