Alejandro H. Morales - cp's OEIS frontend

A357593 Number of faces of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.

8, 26, 88, 298, 1016, 3466, 11832, 40394, 137912, 470858

Offset: 1

Alejandro H. Morales, Oct 05 2022

			For n=1, the polytope is the simplex with vertices (1,0,0), (0,1,0), and (0,0,1) that has a(1)=8 faces (1 empty face, 3 vertices, 3 edges, and 1 facet).

L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Table 3)

Cf. A033303, A007070.

def a(n): return add(PP(n,3,1).f_vector())
def Delta(I,n):
    IM = identity_matrix(n)
    return Polyhedron(vertices=[IM[e] for e in I],backend='normaliz')
def Py(n,SL,yL):
    return sum(yL[i]*Delta(SL[i],n) for i in range(len(SL)))
def PP(n,k,s):
    SS = [set(range(s*i,k+s*i)) for i in range(n)],[1,]*(n)
    return Py(s*(n-1)+k,SS[0],SS[1])
[a(n) for n in range(1,4)]

A357592 Number of edges of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.

Original entry on oeis.org

3, 11, 34, 96, 260, 683, 1757, 4447, 11114, 27493

Offset: 1

Alejandro H. Morales, Oct 05 2022

L. Escobar, P. Gallardo, J. González-Anaya, J. L. González, G. Montúfar, and A. H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022. (See Table 2)

Cf. A033303, A007070.

def a(n): return len(PP(n,3,1).graph().edges())
def Delta(I,n):
    IM = identity_matrix(n)
    return Polyhedron(vertices=[IM[e] for e in I],backend='normaliz')
def Py(n,SL,yL):
    return sum(yL[i]*Delta(SL[i],n) for i in range(len(SL)))
def PP(n,k,s):
    SS = [set(range(s*i,k+s*i)) for i in range(n)],[1,]*(n)
    return Py(s*(n-1)+k,SS[0],SS[1])
[a(n) for n in range(1,4)]

A357484 Number of linearity regions of a max-pooling function with a 3 by n input and 2 by 2 pooling windows.

Original entry on oeis.org

1, 14, 150, 1536, 15594, 158050, 1601356, 16223814, 164366170, 1665216896, 16870539234, 170917714410, 1731590444316, 17542976546494, 177730263461890, 1800609290091936, 18242215773029194, 184814350419581330, 1872379131238643436, 18969325721395559574

Offset: 1

Alejandro H. Morales, Sep 30 2022

a(n) is also the number of vertices of the Minkowski sum of 2*n-2 simplices conv(e_{i,j},e_{i,j+1},e_{i+1,j},e_{i+1,j+1}) for i=0,1 and j=0,...,n-2, viewing R^(3n) having basis {e_{i,j} | i=0,1,2; j=0,...,n-1}.

			For n = 2 the a(2)=14 vertices are (00,10), (00,11), (00,20), (00,21), (01,10), (01,11), (01,20), (01,21), (10,10), (10, 20), (10,21), (11,11), (11, 20), (11, 21), where (ij,kl) represents e_{i,j}+e_{k,l}. The pair (10,11) does not represent vertices since e_{1,0}+e_{1,1} is a convex combination of the vectors 2e_{1,0} + 2e_{1,1}. Ditto for the pair (11,10).

Laura Escobar, Patricio Gallardo, Javier González-Anaya, José L. González, Guido Montúfar, and Alejandro H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (13,-31,20,-4).

Cf. A007070, A033303.

a:= proc(n) option remember;
   if n = 1 then
     return(1);
   elif n = 2 then
     return(14);
   elif n = 3 then
     return(150);
   elif n = 4 then
     return(1536);
   else
     return(13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4));
   end if;
end proc:
seq(a(n), n=1..20);

LinearRecurrence[{13, -31, 20, -4}, {1, 14, 150, 1536}, 20] (* Hugo Pfoertner, Oct 05 2022 *)

@cached_function
def a(n):
    if n < 5: return [1, 14, 150, 1536][n - 1]
    return 13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4)
print([a(n) for n in range(1, 21)])

a(n) = 13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4) for n>= 5.

G.f.: (x+x^2-x^3)/(1-13*x+31*x^2-20*x^3+4*x^4).

A347930 3-Springer numbers.

Original entry on oeis.org

1, 1, 3, 16, 88, 625, 5527, 55760, 640540, 8329326, 120212331, 1905939913, 32987637967, 618591571085, 12489644875037, 270193806214360, 6235154917414954, 152875655211527878, 3968729594485785289, 108754865309750398187, 3137052120203959610759

Offset: 2

Alejandro H. Morales, Sep 19 2021

nonn

a(n) is also the volume of a certain flow polytope.

Arvind Ayyer, Matthieu Josuat-Vergès, and Sanjay Ramassamy, Extensions of partial cyclic orders and consecutive coordinate polytopes, Ann. H. Lebesgue, 3 (2020), 275-297.
R. S. Gonzalez D'Leon, A. H. Morales, C. R. H. Hanusa, and M. Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
S. Ramassamy, Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons, Electron. J. Combin., 25 (2018), #P1.66.

Cf. A001586, A008282, A000111.

wcomps:=proc(n,k)
       option remember;
local ocomps,ncomps,i;
ocomps:=combinat:-composition(n+k,k);
ncomps:={};
for i from 1 to nops(ocomps) do
   ncomps:=ncomps union{[seq(ocomps[i][j]-1,j=1..k)]};
end do;
return [op(ncomps)];
end proc:
b:=proc(s) option remember;
   local k;
   k := nops(s);
   if s = [seq(0,i=1..k)] then
      return(1);
   elif s[1]>0 then
      return(add(b([s[2]+j,op(s[3..k]),s[1]-j-1]),j=0..s[1]-1));
   else
      return(0);
   end if;
end proc:a:=proc(n)   local N,S:   N := n-2;   S := wcomps(N,3);   return add(combinat:-multinomial(N,op(s))*b(s), s in S);end proc:seq(a(n),n=2..10);

a(n) = Sum_{(x,y,z), x+y+z=n-2} ((n-2)!/(x!*y!*z!))*b(x,y,z), where b(x,y,z) are the 3-Entringer numbers defined by Ramassamy.

A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

Original entry on oeis.org

1, 1, 5, 65, 1757, 87129, 7286709, 965911665, 193387756045, 56251615627273, 23021497112124901, 12903943243053179681, 9680994096074346690365, 9530338509606467082850745, 12099590059386455266220499477

Offset: 0

Alejandro H. Morales, Jul 29 2020

nonn

a(n) is also the number of semistandard Young tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+3.

a(n) is also the number of semistandard Young tableaux T of shape (n-1,n-2,...,1) such that j-i < T(i,j) <= n+3 for all cells (i,j).

			For n=2 the a(2)=5 semistandard Young tableaux of skew shape (5,4,3,2,1)/(1) are determined by their first column which are [1,2,3,4], [1,2,3,5], [1,2,4,5], [1,3,4,5], and [2,3,4,5]. Also, the a(2)=5 semistandard Young tableaux of shape (1) with entries between 0 and 5 are [1], [2], [3], [4], and [5]. Also, the a(3)=70-5=65 are the semistandard Young tableaux of shape (2,1) with entries at most 6 excluding the five tableaux whose entry in the first row and first column is 1: [[1,1],[2]], [[1,1],[3]], [[1,1],[4]], and [[1,1],[5]].

A. H. Morales and D. G. Zhu, On the Okounkov-Olshanski formula for standard tableaux of skew shape, arXiv:2007.05006 [math.CO], 2020.

A110501, A005700 gives the number of terms of the Naruse hook length formula for the same skew shape.

b := proc(n)
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n);
end proc:
a := proc(n)
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
end proc:
seq(a(n),n=0..10);

def b(n):
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n) ;
def a(n):
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
[a(i) for i in range(10)]

a(n) = ((2*n+4)!*(2*n+6)!/3!)*(b(n+1)*b(n+3)-b(n+2)^2) where b(n)=A110501(n)/(2*n)!.

A331799 Normalized volume of the Caracol flow polytope. Also equal to the number of "unified diagrams" of the Caracol graph (see Section 4.3 and Section 5 in Benedetti et al. reference).

Original entry on oeis.org

1, 3, 32, 625, 18144, 705894, 34603008, 2051893701, 143000000000, 11464341673642, 1039964049506304, 105353940923859082, 11793014101010071552, 1445828316284179687500, 192713711798795989155840, 27750747808814680091687085, 4293818865468117678192721920

Offset: 1

Alejandro H. Morales, Jan 26 2020

			For n=3, a(3) = 32 = 2*(3+1)^2.

C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, arXiv:1801.07684 [math.CO], 2018-2019.
C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, Trans. Amer. Math. Soc., 372 (2019), 3369-3404.
J. Jang and J. S. Kim, Volumes of flow polytopes related to caracol graphs, arXiv:1911.10703 [math.CO], 2019
M. Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A134264.

a:=proc(n)
  return (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1);
end proc:

Array[(1/#) Binomial[2 # - 2, # - 1] (# + 1)^(# - 1) &, 17] (* Michael De Vlieger, Jan 28 2020 *)

a(n) =  A000108(n-1)*A000272(n+1).
a(n) = (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1).
a(n) = Sum_{i>=0..n-1} binomial(2*n-2,i)*A329057(n-1,i).

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4n-1,4n-2,...,2,1) and mu is the square n^n.

Original entry on oeis.org

1, 16, 4362327818240, 19265181532031090042534736325278852710400, 830325323503973129435791248069702287019820905338483131168940909920954227594481411031040

Offset: 0

Alejandro H. Morales, Sep 05 2017

nonn

The number of standard Young tableaux of a fixed skew shape has a determinantal formula, the Jacobi-Trudi formula. It is rare when a family of skew shapes has a product formula for the number of standard Young tableaux. This product formula has independently been proved using P-Schur functions (by DeWitt) and using the Naruse hook-length formula for skew shapes (by Morales, Pak and Panova).

			a(1)=16 since there are 16 standard Young tableaux of skew shape 321/1 since this is the same as the number of standard Young tableaux of straight shape 321 given by the hook-length formula: 16 = 6!/(3^2*5).

E. A. DeWitt, Identities Relating Schur s-Functions and Q-Functions, Ph.D. thesis, University of Michigan, 2012, 73 pp.
A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.

Cf. A000178, A057863, A008793, A291871, A000085, A061343, A005118.

b:=n->mul(factorial(i),i=1..n-1):
c:=n->mul(doublefactorial(2*i-1),i=1..n-1):
a:=n->factorial(binomial(4*n,2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)):
seq(a(n),n=0..9);

def b(n): return mul([factorial(i) for i in range(1,n)])
def d(n): return factorial(n+1)/(2^((n+1)/2)*factorial((n+1)/2))
def c(n): return mul([d(2*i-1) for i in range(1,n)])
def a(n):
    return factorial(binomial(4*n,2)-n^2)*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n))
[a(n) for n in range(10)]

a(n) = (binomial(4*n,2)-n^2)!*b(n)^3*b(3*n)*c(n)*c(3*n)/(b(2*n)^3*c(2*n)^2*c(4*n)) where b(n) = 1!*2!*...*(n-1)! is the superfactorial A000178(n-1), and c(n) = 1!!*3!!*...*(2*n-3)!! is super doublefactorial A057863(n-1).

a(n) ~ sqrt(Pi) * 3^(9*n^2 - 3*n/2 - 1/24) * 7^(7*n^2 - 2*n + 1/2) * exp(7*n^2/2 - 2*n + 23/56) * n^(7*n^2 - 2*n + 7/8) / (A^(3/2) * 2^(33*n^2 - 6*n - 7/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

A291879 Number of monomials of the Schubert polynomial of the permutation 351624 tensor 1^n.

Original entry on oeis.org

1, 8, 6720, 561120560, 4557185891241984, 3571558033324129373292768, 269111599998006391761541640176800000, 1945556482213500279178010210766074095827609600000, 1347912754604769492992184400055703948513202427323999206349209600

Offset: 0

Alejandro H. Morales, Sep 05 2017

nonn

The permutation 351624 tensor 1^n is the permutation whose permutation matrix is obtained from that of 351624 by replacing each 1 with an n X n identity matrix.

			For n=1 we have that a(1)=8 since the Schubert polynomial of 351624 equals the following sum of eight monomials: x0^3*x1^3*x2 + x0^3*x1^2*x2^2 + x0^2*x1^3*x2^2 + x0^3*x1^3*x3 + x0^3*x1^2*x2*x3 + x0^2*x1^3*x2*x3 + x0^3*x1^2*x3^2 + x0^2*x1^3*x3^2.

A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.
R. P. Stanley, Some Schubert shenanigans, arXiv:1704.00851 [math.CO], 2017.

Cf. A000178, A008793, A291871.

Table[BarnesG[n + 1]^5 * BarnesG[3*n + 1]^2 * BarnesG[5*n + 1] / (BarnesG[2*n + 1]^4 * BarnesG[4*n + 1]^2), {n, 0, 10}] (* Vaclav Kotesovec, Apr 08 2021 *)

b(n) = prod(k=1, n-1, k!);
a(n) = b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2); \\ Michel Marcus, Sep 07 2017

def b(n): return mul([factorial(i) for i in range(1,n)])
def a(n): return b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2)
[a(n) for n in range(10)]

a(n) = b(n)^5*b(3*n)^2*b(5*n)/(b(2*n)^4*b(4*n)^2) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1). [corrected by Vaclav Kotesovec, Apr 08 2021]
a(n) = c(n)*b(3*n)^2*b(6*n)/((7*n^2)!*b(2*n)^2*b(4*n)^2) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1) and c(n) = A291871.
a(n) ~ exp(1/6) * 3^(9*n^2 - 1/6) * 5^(25*n^2/2 - 1/12) / (A^2 * n^(1/6) * 2^(40*n^2 - 2/3)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

A291871 Number of standard Young tableaux of skew shape (3n^(2n), 2*n^n)/(n^n).

Original entry on oeis.org

1, 42, 14617044842400, 5458228515594914179387748450655273588000, 864891828322912925373153355728411014930091519471102108791040960580578545212124160000

Offset: 0

Alejandro H. Morales, Sep 04 2017

nonn

The number of standard Young tableaux of a fixed skew shape has a determinantal formula, the Jacobi-Trudi formula. It is rare when a family of skew shapes has a product formula for the number of standard Young tableaux. This product formula has independently been proved using a combinatorial model for the Selberg integral (by Kim and Oh) and using the Naruse hook-length formula for skew shapes (by Morales, Pak and Panova).

			a(1)=42 since there are 42 standard Young tableaux of skew shape 332/1 since this is the same as the number of standard Young tableaux of straight shape 332 given by the hook-length formula: 42 = 8!/(2^2*3*4^2*5).

J. S. Kim and S. Oh, The Selberg integral and Young books, arXiv:1409.1317 [math.CO], 2014.
J. S. Kim and S. Oh, The Selberg integral and Young books, J. Combin. Theory Ser. A 145 (2017), 1-24.
A. H. Morales, I. Pak, G. Panova, Hook formulas for skew shapes III. Multivariate and product formulas, arXiv:1707.00931 [math.CO], 2017.

Cf. A000178, A008793, A291879, A291908, A000085, A061343.

b:=n->mul(factorial(i),i=1..n-1):
a:=n->factorial(7*n^2)*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n)):
seq(a(i),i=0..9);

b[n_] := Product[i!, {i, n - 1}]; Table[(7 n^2)!*b[n]^5*b[5 n]/(b[2 n]^2*b[6 n]), {n, 0, 4}] (* Michael De Vlieger, Sep 10 2017 *)

def b(n): return mul([factorial(i) for i in range(1,n)])
def a(n): return factorial(7*n^2)*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n))
[a(n) for n in range(10)]

a(n) = (7*n^2)!*b(n)^5*b(5*n)/(b(2*n)^2*b(6*n)) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1).
a(n) = (7*n^2)!*c(n)*b(n)^2*b(2*n)*b(5*n)/(b(6*n)*b(3*n)) where b(n) = 1!*2!*...*(n-1)! is a superfactorial A000178(n-1) and c(n) = A008793.
log a(n) = 7*n^2*log(n) + (75/2 - 15*log(2) - 18*log(3) + 25/2*log(5) + 7*log(7))*n^2 + O(n*log(n)). (See Example 6.2 in Morales et al.)
a(n) ~ sqrt(Pi) * 5^(25*n^2/2 - 1/12) * 7^(7*n^2 + 1/2) * exp(7*n^2/2 + 1/4) * n^(7*n^2 + 3/4) / (A^3 * 2^(22*n^2 - 3/4) * 3^(18*n^2 - 1/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2021

A278289 Number of standard Young tableaux of skew shape (2n-1,2n-2,...,2,1)/(n-1,n-2,..,2,1).

Original entry on oeis.org

1, 1, 16, 101376, 1190156828672, 68978321274090930831360, 40824193474825703180733027309531955200, 440873872874088459550341319780612789503586208384381091840, 140992383930585613207663170866505518985873138480180692888967131590224605582721024

Offset: 0

Alejandro H. Morales, Nov 16 2016

nonn

			For n = 3 there are a(2) = 16 standard tableaux of shape (3,2,1)/(1).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.16.3.

Alejandro H. Morales, Table of n, a(n) for n = 0..22
A. H. Morales, I. Pak and G. Panova, Asymptotics of the number of standard Young tableaux of skew shape, arXiv:1610.07561 [math.CO], 2016; European Journal of Combinatorics, Vol 70 (2018).
A. H. Morales, I. Pak and G. Panova, Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, arXiv:1610.04744 [math.CO], 2016; SIAM Journal of Discrete Mathematics, Vol 31 (2017).
A. H. Morales, I. Pak and M. Tassy, Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings, arXiv:1805.00992 [math.CO], 2018.
A. H. Morales and D. G. Zhu, On the Okounkov--Olshanski formula for standard tableaux of skew shapes, arXiv:2007.05006 [math.CO], 2020.
H. Naruse, Schubert calculus and hook formula, talk slides at 73rd Sém. Lothar. Combin., Strobl, Austria, 2014.
I. Pak, Skew shape asymptotics, a case-based introduction, 2020.
Jay Pantone, File with list of n, a(n) for n = 0..438 (warning: file size is 100MB)

Cf. A005118; for even n the number of terms in Naruse hook length formula is given by A181119 (Corollary 8.1 in arXiv:1610.04744).

a:=proc(k) local lam,mu;
lam:=[seq(2*k-i,i=1..2*k-1)];
mu:=[seq(k-i,i=1..k-1),seq(0,i=1..k)];
factorial(binomial(2*k,2)-binomial(k,2))*LinearAlgebra:-Determinant(Matrix(2*k-1, 2*k-1,(i,j)->`if`(lam[i]-mu[j]-i+j<0,0,1/factorial(lam[i]-mu[j]-i+j))));
end proc:
seq(a(n),n=0..5);

a(n) = ((3*n^2-n)/2)!*det(1/(lambda[i]-mu[j]-i+j)!), where lambda = (2*n-1,2*n-2,...,1) and mu = (n-1,n-2,...,1,0...,0).

There is a constant c such that log(a(k)) = n*log(n)/2 + c*n + o(n) where n = k*(3*k-1)/2 goes to infinity and -0.2368 <= c <= -0.1648. [updated by Alejandro H. Morales, Aug 29 2020]

cp's OEIS Frontend

User: Alejandro H. Morales

A357593 Number of faces of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.

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A357592 Number of edges of the Minkowski sum of n simplices with vertices e_(i+1), e_(i+2), e_(i+3) for i=0,...,n-1, where e_i is a standard basis vector.

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A357484 Number of linearity regions of a max-pooling function with a 3 by n input and 2 by 2 pooling windows.

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A347930 3-Springer numbers.

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A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

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Maple

Sage

Formula

A331799 Normalized volume of the Caracol flow polytope. Also equal to the number of "unified diagrams" of the Caracol graph (see Section 4.3 and Section 5 in Benedetti et al. reference).

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Maple

Mathematica

Formula

A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4*n-1,4*n-2,...,2,1) and mu is the square n^n.

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Maple

Sage

Formula

A291879 Number of monomials of the Schubert polynomial of the permutation 351624 tensor 1^n.

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A291908 Number of standard Young tableaux of skew shape lambda/mu where lambda is the staircase (4n-1,4n-2,...,2,1) and mu is the square n^n.

A291871 Number of standard Young tableaux of skew shape (3n^(2n), 2*n^n)/(n^n).