A001286 -id:A001286 - cp's OEIS frontend

A060614 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=5 and D varies.

0, 1, 14, 1664

Offset: 5

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060616 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=6 and D varies.

Original entry on oeis.org

0, 1, 16, 4032

Offset: 6

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A060619 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=9 and D varies.

Original entry on oeis.org

0, 1, 22, 52224

Offset: 9

Matthieu Latapy (latapy(AT)liafa.jussieu.fr), Apr 13 2001

nonn

			For any Z(d,d), there is a unique tiling therefore the first term of the series is 0. Likewise, there are always two tilings of Z(d+1,d) with a flip between them, therefore the second term of the series is 1.

A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, Oriented Matroids, Encyclopedia of Mathematics 46, Second Edition, Cambridge University Press, 1999.
N. Destainville, R. Mosseri and F. Bailly, Fixed-boundary octagonal random tilings: a combinatorial approach, Journal of Statistical Physics, 102 (2001), no. 1-2, 147-190.
Victor Reiner, The generalized Baues problem, in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996-1997), 293-336, Math. Sci. Res. Inst. Publ., 38, Cambridge Univ. Press, Cambridge, 1999.

M. Latapy, Generalized Integer Partitions, Tilings of Zonotopes and Lattices

Cf. A001286 (case where d=1). Cf. A060595 (number of 3-tilings) for terminology. A diagonal of A060638.

A066118 a(n) = n!(3n-1)/2.

Original entry on oeis.org

1, 5, 24, 132, 840, 6120, 50400, 463680, 4717440, 52617600, 638668800, 8382528000, 118313395200, 1787154969600, 28768836096000, 491685562368000, 8892185702400000, 169662903201792000, 3406062811447296000, 71770609241210880000, 1583819207322992640000

Offset: 1

George E. Antoniou, Dec 05 2001

Harry J. Smith, Table of n, a(n) for n=1..100

Array[#!*(3*#-1)/2 &, 25] (* Paolo Xausa, Feb 16 2024 *)

{ for (n=1, 100, write("b066118.txt", n, " ", (n!*(3*n - 1))/2) ) } \\ Harry J. Smith, Feb 01 2010

E.g.f.: (1+x/2)/(1-x)^2. - Len Smiley, Dec 06 2001
3*a(n) +(-3*n-7)*a(n-1) +4*(n-1)*a(n-2)=0. - R. J. Mathar, Oct 30 2015
(-3*n+4)*a(n) +n*(3*n-1)*a(n-1)=0. - R. J. Mathar, Oct 30 2015
a(n) = A001563(n) + A001286(n). - Anton Zakharov, Oct 17 2016

A139359 Number L([n],m) of ways the labeled parts of each integer partition of n can be distributed into m nonempty labeled boxes.

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 5, 16, 36, 24, 7, 46, 150, 240, 120, 11, 114, 546, 1560, 1800, 720, 15, 614, 2058, 8400, 16800, 15120, 5040, 22, 1366, 6984, 40848, 126000, 191520, 141120, 40320, 30, 12516, 73488, 192816, 834120, 1905120, 2328480, 1451520, 362880

Offset: 1

Thomas Wieder, Apr 14 2008

nonn

This formula is related to a formula given by Riordan, see Riordan, 1958, page 94. Furthermore, this formula is related to the distribution of labeled elements into labeled boxes, as described by A019538.
The first column is equal to A000041 = number of partitions of n (the partition numbers).
The main diagonal is equal to the A000142 = Factorial numbers: n!
The second diagonal is equal to A001286 = Lah numbers: (n-1)*n!/2.
The third diagonal is equal to A019538 = Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
If we normalize the m-th column by m! we get the triangle
  1
  2 1
  3 3 1
  5 8 6 1
  7 23 25 10 1
  11 57 91 65 15 1
  15 307 343 350 140 21 1
  22 683 1164 1702 1050 266 28 1
  30 6258 12248 8034 6951 2646 462 36 1
In this triangle we observe:
The second diagonal is equal to A000217 = Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
The third diagonal is composed of numbers belonging to A095660 = Pascal (1,3) triangle.

			Triangle begins:
  1
  2 2
  3 6 6
  5 16 36 24
  7 46 150 240 120
  11 114 546 1560 1800 720
  15 614 2058 8400 16800 15120 5040
  22 1366 6984 40848 126000 191520 141120 40320
  30 12516 73488 192816 834120 1905120 2328480 1451520 362880
  ...

John Riordan: Introduction to Combinatorics, John Wiley & Sons, New York, 1958, ISBN 0-486-42536-3.

Thomas Wieder, Further comments on this sequence

Cf. A019538, A137383.

A142706 Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).

Original entry on oeis.org

1, 4, 2, 11, 22, 3, 26, 132, 78, 4, 57, 604, 906, 228, 5, 120, 2382, 7248, 4764, 600, 6, 247, 8586, 46857, 62476, 21465, 1482, 7, 502, 29216, 264702, 624760, 441170, 87648, 3514, 8, 1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 24 2008

			Triangle T(n, k) starts:
{   1};
{   4,     2};
{  11,    22,       3};
{  26,   132,      78,       4};
{  57,   604,     906,     228,       5};
{ 120,  2382,    7248,    4764,     600,       6};
{ 247,  8586,   46857,   62476,   21465,    1482,      7};
{ 502, 29216,  264702,  624760,  441170,   87648,   3514,   8};
{1013, 95680, 1365576, 5241416, 6551770, 2731152, 334880, 8104, 9}.

Cf. A173018, A001286 (row sums).

T := (n, k) -> k * combinat:-eulerian1(n+1, k):
for n from 1 to 9 do seq(T(n, k), k = 1..n) od; # Peter Luschny, Feb 07 2023

T[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
Table[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], {n, 1, 10}];
Table[CoefficientList[D[Sum[T[n, k]*x^k, {k, 0, n - 1}], x], x], {n, 1, 10}];
Flatten[%]
(* Alternative: *) Needs["Combinatorica`"]
Flatten[Table[k*Eulerian[n+1, k], {n, 1, 9}, {k, 1, n}]] (* Peter Luschny, Feb 07 2023 *)

Let E(n, x) = Sum_{j=0..k} A173018(n, k)*x^k and E'(n, x) = (d/dx) E(x, n). Then T(n, k) = [x^(k-1)] E'(n+1, x).

Edited by Peter Luschny, Feb 07 2023

A176860 Triangle, read by rows, T(n, k) = (-1)^k * (n-k+1)^(n+2) * binomial(n+1, k).

Original entry on oeis.org

1, 8, -2, 81, -48, 3, 1024, -972, 192, -4, 15625, -20480, 7290, -640, 5, 279936, -468750, 245760, -43740, 1920, -6, 5764801, -11757312, 8203125, -2293760, 229635, -5376, 7, 134217728, -322828856, 282175488, -109375000, 18350080, -1102248, 14336, -8

Offset: 0

Roger L. Bagula, Apr 27 2010

			Triangle begins as:
          1;
          8,         -2;
         81,        -48,         3;
       1024,       -972,       192,         -4;
      15625,     -20480,      7290,       -640,        5;
     279936,    -468750,    245760,     -43740,     1920,       -6;
    5764801,  -11757312,   8203125,   -2293760,   229635,    -5376,     7;
  134217728, -322828856, 282175488, -109375000, 18350080, -1102248, 14336, -8;

F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 267.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A001286.

[(-1)^k*(n-k+1)^(n+2)*Binomial(n+1,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

T[n_, k_]:= (-1)^k*(n-k+1)^(n+2)*Binomial[n+1, k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

flatten([[ (-1)^k*(n-k+1)^(n+2)*binomial(n+1,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021

T(n, k) = (-1)^k * (n-k+1)^(n+2) * binomial(n+1, k).

Sum_{k=0..n} T(n, k) = (n + 1)*(n + 2)!/2 = A001286(n+2). - G. C. Greubel, Feb 07 2021

Edited by G. C. Greubel, Feb 07 2021

A178232 A triangle sequence derived from setting an Euler numbers A122045 generalization equal to the Eulerian numbers A008292 to get a generating function expansion: p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(tx) + t - exp(t) x)).

Original entry on oeis.org

0, 0, 1, 6, 1, 1, 36, 8, 3, 7, 1, 240, 60, -20, 81, 11, 21, 1, 1800, 480, -510, 822, 143, 173, 123, 51, 1, 15120, 4200, -7560, 8526, 2450, 239, 2381, 435, 715, 113, 1, 141120, 40320, -102480, 93744, 43512, -21320, 36991, 2943, 11035, 4035, 3139, 239, 1

Offset: 0

Roger L. Bagula, May 23 2010

The first column gives the Lah numbers A001286: (n - 1)*n!/2;
{0,0,1, 6, 36, 240, 1800, 15120, 141120, 1451520, ...}.
Row sums are {0, 0, 1, 8, 55, 394, 3083, 26620, 253279, 2642390, 30052699, ...}.
The equation solved in the integer q was
  q*exp(x*t)/(q - 1 + exp(t)) - (1 - t)/(1 - t*exp(x*(1 - t))) = 0.
Factors and the n! first term from taken out in Mathematica to give a more simple set of coefficients.
The idea in solving for an integer q here is to get a polynomial that behaves as a generalization of both types.
No q-form value for q=n=0,1 is expected.

			{0},
{0},
{1},
{6, 1, 1},
{36, 8, 3, 7, 1},
{240, 60, -20, 81, 11, 21, 1},
{1800, 480, -510, 822, 143, 173, 123, 51, 1},
{15120, 4200, -7560, 8526, 2450, 239, 2381, 435, 715, 113, 1},
{141120, 40320, -102480, 93744, 43512, -21320, 36991, 2943, 11035, 4035, 3139, 239, 1},
{1451520, 423360, -1391040, 1103760, 763056, -585432, 527544, 71353, 82513, 107377, 39589, 36349, 11947, 493, 1},
{16329600, 4838400, -19504800, 13940640, 13361040, -12088080, 7137270, 2643650, -749001, 2527719, 165459, 900099, 256743, 251073, 41883, 1003, 1}

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 78-79.
L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245.

L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.

Cf. A008292, A122045, A156222.

p[t_] = ((-1 + Exp[x]) (-1 + x)/(-1 + Exp[t*x] + t - Exp[t]* x));
a = Table[ CoefficientList[FullSimplify[ExpandAll[(FullSimplify[ExpandAll[ -(1/((-1 + Exp[x])*(-1 + x)))*x^(n + 1)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]] - n!)/(x^2*(-1 + x))]], x], {n, 0, 10}] Flatten[a]

p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(t*x) + t - exp(t)* x)).

A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

Original entry on oeis.org

1, 6, 3, 36, 24, 12, 240, 180, 120, 60, 1800, 1440, 1080, 720, 360, 15120, 12600, 10080, 7560, 5040, 2520, 141120, 120960, 100800, 80640, 60480, 40320, 20160, 1451520, 1270080, 1088640, 907200, 725760, 544320, 362880, 181440, 16329600, 14515200, 12700800, 10886400, 9072000, 7257600, 5443200, 3628800, 1814400

Offset: 2

Geoffrey Critzer, Jan 09 2013

Row sums = A001809.

Column for k = 0 is A001286.

			T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
Triangle T(n,k) begins:
       1;
       6,      3;
      36,     24,     12;
     240,    180,    120,    60;
    1800,   1440,   1080,   720,   360;
   15120,  12600,  10080,  7560,  5040,  2520;
  141120, 120960, 100800, 80640, 60480, 40320, 20160;
  ...

Alois P. Heinz, Rows n = 2..142, flattened

Cf. A001286, A001809, A055303.

nn=10;Range[0,nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x),{x,0,nn}],{x,y}]//Grid

E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).

A317483 Circuit rank of the n-Bruhat graph.

Original entry on oeis.org

0, 0, 1, 13, 121, 1081, 10081, 100801, 1088641, 12700801, 159667201, 2155507201, 31135104001, 479480601601, 7846046208001, 135998134272001, 2489811996672001, 48017802792960001, 973160803270656001, 20679667069501440001, 459818479545384960001, 10678006913887272960001

Offset: 1

Eric W. Weisstein, Jul 29 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Bruhat Graph
Eric Weisstein's World of Mathematics, Circuit Rank

Cf. A000142, A001286.

Table[(n - 3) n!/2 + 1, {n, 20}] (* Eric W. Weisstein, May 10 2019 *)

a(n)={n!*(n-1)\2 - n! + 1} \\ Andrew Howroyd, Jul 30 2018

From Andrew Howroyd, Jul 30 2018: (Start)
a(n) = A001286(n) - A000142(n) + 1.
a(n) = (n-3)*n!/2 + 1. (End)

Terms a(7) and beyond from Andrew Howroyd, Jul 30 2018

cp's OEIS Frontend

A060614 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=5 and D varies.

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A060616 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=6 and D varies.

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A060619 Number of flips between the d-dimensional tilings of the unary zonotope Z(D,d). Here d=9 and D varies.

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A066118 a(n) = n!*(3*n-1)/2.

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Mathematica

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A139359 Number L([n],m) of ways the labeled parts of each integer partition of n can be distributed into m nonempty labeled boxes.

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A142706 Coefficients of the derivatives of the Eulerian polynomials (with indexing as in A173018).

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Maple

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A176860 Triangle, read by rows, T(n, k) = (-1)^k * (n-k+1)^(n+2) * binomial(n+1, k).

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Magma

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A178232 A triangle sequence derived from setting an Euler numbers A122045 generalization equal to the Eulerian numbers A008292 to get a generating function expansion: p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(t*x) + t - exp(t)* x)).

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A066118 a(n) = n!(3n-1)/2.

A178232 A triangle sequence derived from setting an Euler numbers A122045 generalization equal to the Eulerian numbers A008292 to get a generating function expansion: p(x,t) = ((-1 + exp(x)) (-1 + x)/(-1 + exp(tx) + t - exp(t) x)).