A007724 -id:A007724 - cp's OEIS frontend

A003121 Strict sense ballot numbers: n candidates, k-th candidate gets k votes.

1, 1, 1, 2, 12, 286, 33592, 23178480, 108995910720, 3973186258569120, 1257987096462161167200, 3830793890438041335187545600, 123051391839834932169117010215648000, 45367448380314462649742951646437285434233600, 207515126854334868747300581954534054343817468395494400

Offset: 0

Colin Mallows

Also, number of even minus number of odd extensions of truncated (n-1) X n grid diagram.
Also, a(n) is the degree of the spinor variety, the complex projective variety SO(2n+1)/U(n). See Hiller's paper. - Burt Totaro (b.totaro(AT)dpmms.cam.ac.uk), Oct 29 2002
Also, number of ways of placing 1, ..., n*(n+1)/2 in a triangular array such that both rows and columns are increasing, i.e., the number of shifted standard Young tableaux of shape (n, n-1, ..., 1).
E.g., a(3) = 2 as we can write:
  1            1
  2 3    or    2 4
  4 5 6        3 5 6
(or transpose these to have shifted tableaux). - Jon Perry, Jun 13 2003, edited by Joel B. Lewis, Aug 27 2011
Also, the number of symbolic sequences on the n symbols 0,1, ..., n-1. A symbolic sequence is a sequence that has n occurrences of 0, n-1 occurrences of 1, ..., one occurrence of n-1 and satisfies the condition that between any two consecutive occurrences of the symbol i it has exactly one occurrence of the symbol i+1. For example, the two symbolic sequences on 3 symbols are 012010 and 010210. The Shapiro-Shapiro paper shows how such sequences arise in the study of the arrangement of the real roots of a polynomial and its derivatives. There is a natural bijection between symbolic sequences and the triangular arrays described above. - Peter Bala, Jul 18 2007
a(n) also appears to be the number of chains from w_0 down to 1 in a certain suborder of the strong Bruhat order on S_n: we let v cover (ij)*v only if i,j straddle the leftmost descent in v. For n=3 and drawing that descent with a |, this order is 3|21 > 23|1 > 13|2 & 2|13 > 123, with two maximal chains. - Allen Knutson (allenk(AT)math.cornell.edu), Oct 13 2008
Number of ways to arrange the numbers 1,2,...,n(n+1)/2 in a triangle so that the rows interlace; e.g., one of the 12 triangles counted by a(4) is
        6
      4   8
    2   5   9
  1   3   7   10
- Clark Kimberling, Mar 25 2012
Also, the number of maps from n X n pipe dreams (rc-graphs) to words of adjacent transpositions in S_n that send a crossing of pipes x and y in square (i,j) to the transposition (i+j-1,i+j) swapping x and y. Taking the pictorial image of a permutation as a wiring diagram, these are maps from pipe dreams to wiring diagrams that send crossings of pipes to crossings of similarly labeled wires. - Cameron Marcott, Nov 26 2012
Number of words of length T(n)=n*(n+1)/2 with n 1's, (n-1) 2's, ..., and 1 n such that counting the numbers from left to right we always have |1| > |2| > |3| > ... > |n|. The 12 words for n=4 are 1111222334, 1111223234, 1112122334, 1112123234, 1112212334, 1112213234, 1112231234, 1121122334, 1121123234, 1121212334, 1121213234 and 1121231234. - Jon Perry, Jan 27 2013
Regarding the comment dated Mar 25 2012, it is assumed that each row is in increasing order, as in the example dated Jul 12 2012. How many row-interlacing triangles are there without that restriction? - Clark Kimberling, Dec 02 2014
Number of maximal chains of length binomial(n+1,2) in the Tamari lattice T_{n+1}. For n=2 there is 1 maximal chain of length 3 in the Tamari lattice T_3. - Alois P. Heinz, Dec 04 2015
The normalized volume of the subpolytope of the Birkhoff polytope obtained by taking the convex hull of all n X n permutation matrices corresponding to permutations that avoid the patterns 132 and 312. - Robert Davis, Dec 04 2016
From Emily Gunawan, Feb 26 2022: (Start)
The number of maximal chains in the lattice of permutations which avoid both the patterns 132 and 312, as a sublattice of the weak order on the symmetric group. For example, there are exactly 12 maximal chains in the sublattice for the weak order on the symmetric group on 5 elements.
The number of words in the commutation class of the c-sorting word of the longest permutation w_0 in the symmetric group for the Coxeter element c = s_1 s_2 s_3 s_4 s_5 ... . For example, the c-sorting word of w_0 for s_1 s_2 s_3 s_4 is the reduced word s_1 s_2 s_3 s_4 | s_1 s_2 s_3 | s_1 s_2 | s_1, and there are exactly 12 words in its commutation class.
The number of maximal chains in the lattice of c-singletons for the symmetric group, for the Coxeter element c = s_1 s_2 s_3 s_4 s_5 ... . For example, there are exactly 12 maximal chains in the lattice of c-singletons for c = s_1 s_2 s_3 s_4. (End)

			From _R. H. Hardin_, Jul 06 2012: (Start)
The a(4) = 12 ways to fill a triangle with the numbers 0 through 9:
.
     5         6         6         5
    3 7       3 7       2 7       2 7
   1 4 8     1 4 8     1 4 8     1 4 8
  0 2 6 9   0 2 5 9   0 3 5 9   0 3 6 9
.
     5         3         3         4
    3 6       2 6       2 7       3 7
   1 4 8     1 5 8     1 5 8     1 5 8
  0 2 7 9   0 4 7 9   0 4 6 9   0 2 6 9
.
     4         4         5         4
    2 6       2 7       2 6       3 6
   1 5 8     1 5 8     1 4 8     1 5 8
  0 3 7 9   0 3 6 9   0 3 7 9   0 2 7 9
.
(End)

G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..30
E. Aas and S. Linusson, Continuous multiline queues and TASEP, 2014; also arxiv version, arXiv:1501.04417 [math.CO], 2015-2017.
Joerg Arndt, The a(5)=286 Young tableaux of shape [1,2,3,4,5].
D. E. Barton and C. L. Mallows, Some aspects of the random sequence, Ann. Math. Statist. 36 (1965) 236-260.
Andrew Beveridge, Ian Calaway, and Kristin Heysse, de Finetti Lattices and Magog Triangles, arXiv:1912.12319 [math.CO], 2019.
R. Davis and B. Sagan, Pattern-Avoiding Polytopes, arXiv:1609.01782 [math.CO], 2016.
William T. Dugan, Maura Hegarty, Alejandro H. Morales, and Annie Raymond, Generalized Pitman-Stanley flow polytopes, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #80.
William T. Dugan, Maura Hegarty, Alejandro H. Morales, and Annie Raymond, Generalized Pitman-Stanley polytope: vertices and faces, arXiv:2307.09925 [math.CO], 2023.
S. Fishel and L. Nelson, Chains of maximum length in the Tamari lattice, Proc. Amer. Math. Soc. 142 (2014), 3343-3353.
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
H. Hiller, Combinatorics and intersection of Schubert varieties, Comment. Math. Helv. 57 (1982), 41-59.
C. Hohlweg, C. Lange, and H. Thomas, Permutahedra and generalized associahedra, arXiv:0709.4241 [math.CO], 2007-2008; Adv. Math. 226 (2011), no. 1, 608-640.
G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]
Joshua Maglione and Christopher Voll, Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration, arXiv:2410.08075 [math.CO], 2024. See pp. 30, 39.
Colin Mallows, Letter to N. J. A. Sloane, date unknown.
Svante Linusson and Samu Potka, New properties of the Edelman-Greene bijection, arXiv:1804.10034 [math.CO], 2018.
N. Reading, Cambrian lattices, arXiv:math/0402086 [math.CO], 2004-2005; Adv. Math. 205 (2006), no. 2, 313-353.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
B. Shapiro and M. Shapiro, A few riddles behind Rolle's theorem, arXiv:math/0302215 [math.CA], 2003-2005; Amer. Math. Monthly, 119 (2012), 787-795.
George Story, Counting Maximal Chains in Weighted Voting Posets, Rose-Hulman Undergraduate Mathematics Journal, Vol. 14, No. 1, 2013.
R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.
Dennis White, Sign-balanced posets, Journal of Combinatorial Theory, Series A, Volume 95, Issue 1, July 2001, Pages 1-38.
Wikipedia, Tamari lattice

Cf. A005118, A018241, A007724, A004065, A131811, A064049, A064050.
A213457 is also closely related.
Cf. A000108, A027686, A282698.

f:= n-> ((n*n+n)/2)!*mul((i-1)!/(2*i-1)!, i=1..n); seq(f(n), n=0..20);

f[n_] := (n (n + 1)/2)! Product[(i - 1)!/(2 i - 1)!, {i, n}]; Array[f, 14, 0] (* Robert G. Wilson v, May 14 2013 *)

a(n)=((n*n+n)/2)!*prod(i=1,n,(i-1)!/(2*i-1)!)

a(n) = binomial(n+1, 2)!*(1!*2!*...*(n-1)!)/(1!*3!*...*(2n-1)!).

a(n) ~ sqrt(Pi) * exp(n^2/4 + n/2 + 7/24) * n^(n^2/2 + n/2 + 23/24) / (A^(1/2) * 2^(3*n^2/2 + n + 5/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

More terms from Michael Somos
Additional references from Frank Ruskey
Formula corrected by Eric Rowland, Jun 18 2010

A217800 Number of alternating permutations on 2n+1 letters that avoid a certain pattern of length 4 (see Lewis, 2012, Appendix, for precise definition).

Original entry on oeis.org

1, 2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100

Offset: 0

N. J. A. Sloane, Oct 12 2012

nonn

1 together with A007724. - Omar E. Pol, Aug 22 2016

K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.

Cf. A007724, A181197, A217799-A217830, A005789, A006480, A007724.

[Factorial(3*n+3)/((4*(n+1)^2-1)*Factorial((n+1))^2*Factorial(n+ 2)): n in [0..20]]; // Vincenzo Librandi, Aug 30 2014

a := n -> (-1)^n*hypergeom([-2-2*n, -2*n, -2*n-1], [2, 3], 1):
seq(round(evalf(a(n), 32)), n=0..20); # Peter Luschny, Aug 29 2014

Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 0, 20}] (* Michael De Vlieger, Aug 22 2016 *)

a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014

From Karol A. Penson, Aug 10 2014: (Start)
O.g.f.(in Maple notation): hypergeom([1/2, 1, 4/3, 5/3], [2, 5/2, 3], 27*z);a(n) ~ (1/93312)*sqrt(3)*27^n*(314928*n^4-1644624*n^3+5545260*n^2 -15387660*n+38310503)/(Pi*n^8), for n -> infinity.
Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:
a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where
W(x) = 1/(Pi*sqrt(x))+sqrt(x)/Pi-(9/20)*sqrt(3)*2^(1/3)* hypergeom([-2/3, -1/6, 1/3], [2/3, 11/6], (1/27)*x)*x^(1/3)/ (sqrt(Pi)*Gamma(5/6)*Gamma(2/3))-(27/56)*2^(2/3)*Gamma(5/6) *Gamma(2/3)*hypergeom([-1/3, 1/6, 2/3], [4/3, 13/6], (1/27)*x)* x^(2/3)/Pi^(5/2).
W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.
This is the solution of the Hausdorff moment problem and is unique.
a(n) = (1/2)*(n+3)!/((4*(n+1)^2-1)*(n+1)!)*A005789(n), where A005789(n) are the three-dimensional Catalan numbers (see the Gorska and Penson link).(End)
a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - Peter Luschny, Aug 15 2014
a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - Peter Luschny, Aug 29 2014
(2*n+3)*(n+2)*(n+1)*a(n) -3*(3*n+2)*(2*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3^(3*n + 7/2) / (8*Pi*n^4). - Vaclav Kotesovec, Jun 09 2019

More terms from Alois P. Heinz, Aug 22 2016

Merged with A241958 by R. J. Mathar, Jul 07 2023

A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).

Original entry on oeis.org

1, 1, 3, 13, 69, 417, 2763, 19609, 146793, 1146833, 9278595, 77292261, 659973933, 5756169681, 51137399979, 461691066417, 4228199347281, 39216540096993, 367890444302787, 3486697883136957, 33353178454762389, 321754445379041601, 3127955713554766923, 30624486778208481993, 301790556354721667769, 2991957347531210976817

Offset: 0

Joerg Arndt, Jan 04 2024

Cf. A001181, A007724, A056939, A217800, A359363, A368709.

seq(simplify( hypergeom([-1 - n, -n, 1 - n], [2, 3], -2) ), n = 0..25); # Peter Bala, Sep 09 2024

Table[HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -2], {n, 0, 30}] (* Vaclav Kotesovec, Jan 04 2024 *)
a[0] := 1; a[n_] := 2^(n + 1)/(n*(n + 1)^2)*Sum[(1/2)^k*Binomial[n + 1, k - 1]*Binomial[n + 1, k]*Binomial[n + 1, k + 1], {k, 1, n}]; Table[a[n], {n, 0, 25}] (* Detlef Meya, May 28 2024 *)

def A368708(n):
    if n == 0: return 1
    return sum(2**k * v for k, v in enumerate(A359363Row(n))) // 2
print([A368708(n) for n in range(26)]) # Peter Luschny, Jan 04 2024

def A368708(n): return PolyA359363(n, 2) // 2 if n > 0 else 1
print([A368708(n) for n in range(23)])  # Peter Luschny, Jan 04 2024

a(n) = (1/2)*B(n, 2) where B(n, x) are the Baxter polynomials with coefficients A359363, for n > 0. - Peter Luschny, Jan 04 2024
a(n) ~ 3^(n + 7/6) * (2^(2/3) + 2^(1/3) + 1)^(n + 5/3) / (2^(4/3) * Pi * n^4). - Vaclav Kotesovec, Jan 04 2024
a(0) = 1, a(n) = 2^(n + 1)/(n*(n + 1)^2)*Sum_{k=1..n} (1/2)^k*binomial(n + 1, k - 1)*binomial(n + 1, k)*binomial(n + 1, k + 1). - Detlef Meya, May 29 2024
From Peter Bala, Sep 09 2024: (Start)
a(n+1) = Sum_{k = 0..n} A056939(n, k)*2^k.
P-recursive: (n+1)*(n+3)*(n+2)*(3*n-2)*a(n) = 3*(9*n^3+3*n^2-4*n+4)*(n+1)*a(n-1) + 3*(n-2)*(3*n-1)*(9*n^2-3*n-10)*a(n-2) + 27*(3*n+1)*(n-3)*(n-2)^2*a(n-3) = 0 with a(0) = 1, a(1) = 1 and a(2) = 3. (End)

A241958 Duplicate of A217800.

Original entry on oeis.org

Offset: 0

Karol A. Penson, Aug 10 2014

dead

This is a duplicate of A217800 or of A007724. - Alois P. Heinz, Aug 22 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008, 2013

[Factorial(3*n+3)/((4*(n+1)^2-1)*Factorial((n+1))^2*Factorial(n+ 2)): n in [0..20]]; // Vincenzo Librandi, Aug 30 2014

a := n -> (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2, 3],1):
seq(round(evalf(a(n),32)),n=0..20); # Peter Luschny, Aug 29 2014

Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3},
1], {n, 0, 20}] (* Michael De Vlieger, Aug 22 2016 *)

a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014

O.g.f.(in Maple notation): hypergeom([1/2, 1, 4/3, 5/3], [2, 5/2, 3], 27*z);
a(n) ~ (1/93312)*sqrt(3)*27^n*(314928*n^4-1644624*n^3+5545260*n^2 -15387660*n+38310503)/(Pi*n^8), for n -> infinity.
Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:
a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where
W(x) = 1/(Pi*sqrt(x))+sqrt(x)/Pi-(9/20)*sqrt(3)*2^(1/3)* hypergeom([-2/3, -1/6, 1/3], [2/3, 11/6], (1/27)*x)*x^(1/3)/ (sqrt(Pi)*Gamma(5/6)*Gamma(2/3))-(27/56)*2^(2/3)*Gamma(5/6) *Gamma(2/3)*hypergeom([-1/3, 1/6, 2/3], [4/3, 13/6], (1/27)*x)* x^(2/3)/Pi^(5/2).
W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.
This is the solution of the Hausdorff moment problem and is unique.
a(n) = (1/2)*(n+3)!/((4*(n+1)^2-1)*(n+1)!)*A005789(n), where A005789(n) are the three-dimensional Catalan numbers (see the Gorska and Penson link).
a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - Peter Luschny, Aug 15 2014
a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - Peter Luschny, Aug 29 2014
(2*n+3)*(n+2)*(n+1)*a(n) -3*(3*n+2)*(2*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Jun 14 2016

cp's OEIS Frontend

A003121 Strict sense ballot numbers: n candidates, k-th candidate gets k votes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A217800 Number of alternating permutations on 2n+1 letters that avoid a certain pattern of length 4 (see Lewis, 2012, Appendix, for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A368708 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -2).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

A241958 Duplicate of A217800.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A368709 a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], +2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

SageMath

Formula