A005802 -id:A005802 - cp's OEIS frontend

A284712 Number of indecomposable permutations avoiding the pattern 4321.

1, 1, 3, 12, 56, 289, 1603, 9391, 57437, 363773, 2371551, 15842850, 108070370, 750655459, 5297314241, 37909476912, 274693258884, 2012767854337, 14897269924571, 111269729060260, 838012360130944, 6359434553977763, 48597188639038769, 373757879279966501

Offset: 1

N. J. A. Sloane, Apr 02 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
A. L. L. Gao, S. Kitaev, P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.

Cf. A005802, A022558.

Rest[CoefficientList[Series[1 - 6*x^2/(1 + 5*x - (1-9*x)^(3/4) * (1-x)^(1/4) * HypergeometricPFQ[{-1/4, 3/4}, {1}, 64*x/((x-1)*(1-9*x)^3)]), {x, 0, 30}], x]] (* Vaclav Kotesovec, Apr 03 2017 *)

a(n) ~ c * 3^(2*n) / n^4, where c = 2.102569600767282153... . - Vaclav Kotesovec, Apr 03 2017

More terms from Vaclav Kotesovec, Apr 03 2017

A129130 Number of triples of standard tableaux with the same shape of height less than or equal to three.

Original entry on oeis.org

1, 1, 2, 10, 63, 531, 6201, 70477, 897149, 12772405, 188334604, 2939523104, 47902337803, 809518276503, 14134544880444, 252955559204532, 4651455689358657, 87356706437180529, 1669767921758484702

Offset: 0

Mike Zabrocki, Mar 30 2007

nonn

			f_111 = f_3 = 1, f_21 = 2 therefore a(3) = f_111^3 + f_21^3 + f_3^3 = 1+8+1 = 10

F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.

Cf. A001006, A005802, A003161.

sum_lambda f_lambda^3 where the sum is over lambda partitions of length less than or equal to 3 and f_lambda is the number of standard tableaux of shape lambda

A145845 Number of permutations of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.

Original entry on oeis.org

1, 2, 7, 34, 208, 1504, 12283, 109778, 1050820, 10614856, 111978128, 1224261856, 13792583296, 159411938560, 1883550536707, 22687603653106, 277940485660012, 3456490397570392, 43565433620294908, 555752354850506312, 7167182317486700416, 93348781597357983232, 1226830676118851157712

Offset: 0

Eric S. Egge, Oct 21 2008

nonn

Table[Sum[ Binomial[n, j]^2*(1/((j + 1)^2*(j + 2)))* Sum[Binomial[2*i, i]*Binomial[j + 1, i + 1]* Binomial[j + 2, i + 1], {i, 0, j}], {j, 0, n}], {n, 0, 20}]

/* using formula given; this gives fractions! */
C=binomial;
a(n)=sum(j=0, n, C(n,j)^2 * (1/((j+1)^2*(j+2))) * sum(i=0, j, C(2*i,i)*C(j+1,i+i)*C(j+2,i+1)));
\\ Joerg Arndt, Feb 18 2015

/* Using a(n) = sum(j=0, n, C(n,j)^2 * A005802(j)). */
f(n)= 2 * sum(k=0,n, binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2+2*k+1-n-2*k*n)/((k+1)^2 * (k+2) * (n-k+1)));
vector(33, N, my(n=N-1); sum(j=0,n, f(j) * C(n,j)^2 ) )
\\ Joerg Arndt, Feb 18 2015

a(n) = sum(j=0, n, C(n,j)^2 * A005802(j)).
a(n) = sum(j=0, n, C(n,j)^2 * (1/((j+1)^2 (j+2))) * sum(i=0, j, C(2*i,i) * C(j+1,i+i) * C(j+2,i+1))) where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)^3*(3*n+1)*a(n) = 2*(30*n^4 + 67*n^3 + 29*n^2 - 10*n - 8)*a(n-1) - 64*(n-1)^2*n*(3*n+4)*a(n-2). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 2^(4*n+5) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Feb 18 2015

Added more terms, Joerg Arndt, Feb 18 2015

A145846 Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 6.

Original entry on oeis.org

1, 2, 8, 47, 357, 3270, 34515, 406460, 5215829, 71677058, 1041363040, 15841778155, 250494079945, 4093630537014, 68830515423498, 1186424966652225, 20902566718237725, 375485138838707850, 6863181435514906992, 127420716337372828539, 2399321143670605041105

Offset: 0

Eric S. Egge, Oct 21 2008

nonn

Table[Sum[ Binomial[n, j]^2*((1/(n - j + 1))* Binomial[2*(n - j), n - j]/((j + 1)^2*(j + 2)))* Sum[Binomial[2*i, i]*Binomial[j + 1, i + 1]* Binomial[j + 2, i + 1], {i, 0, j}], {j, 0, n}], {n, 0, 20}]

a(n) = sum(j, 0, n, C(n,j)^2 * A000108(n-j) * A005802(j)), where C(n,j) = n!/(j!(n-j)!).
Recurrence: (n+2)^2*(n+3)^2*(64*n^3 + 96*n^2 - 36*n - 79)*a(n) = (2240*n^7 + 13664*n^6 + 26068*n^5 + 7303*n^4 - 27638*n^3 - 20581*n^2 + 5964*n + 5940)*a(n-1) - (n-1)^2*(16576*n^5 + 61344*n^4 + 25556*n^3 - 84501*n^2 - 46860*n - 15300)*a(n-2) + 225*(n-2)^2*(n-1)^2*(64*n^3 + 288*n^2 + 348*n + 45)*a(n-3). - Vaclav Kotesovec, Feb 18 2015
a(n) ~ 5^(2*n+13/2) / (128 * Pi^2 * n^6). - Vaclav Kotesovec, Feb 18 2015

More terms from Vaclav Kotesovec, Feb 18 2015

A224298 Number of permutations in S_n containing exactly 3 increasing subsequences of length 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 10, 196, 2477, 25886, 244233, 2167834, 18510734, 154082218, 1260811144, 10198142484, 81848366557, 653537296202, 5201485318177, 41321901094750, 327996498249202

Offset: 1

Brian Nakamura, Apr 03 2013

nonn

B. Nakamura and D. Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes, Adv. in Appl. Math. 50 (2013), 356-366.

B. Nakamura and D. Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes

Cf. A005802, A217057, A224249.

# programs can be obtained from the Nakamura and Zeilberger link.

A247591 Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.

Original entry on oeis.org

1, 1, 3, 16, 126, 1296, 16071, 228514, 3607890, 61891050, 1135871490, 22049362440, 448790912004, 9512960347260, 208858963314735, 4728736078065810, 110006925920592810, 2621619942885055530, 63840054782606886630, 1585094577104979776880, 40054740803371374834780, 1028483346608802276173280

Offset: 0

Paul Zinn-Justin, Sep 20 2014

nonn

The 6-dimensional representation is the usual representation of SO(6)=A_3.

			For 2n=6, there are 15 invariants corresponding to all ways of pairing the 6 indices with the metric tensor, plus one invariant which is the completely skew-symmetric 6-index tensor.

Cf. A095922, A059710.

a[0] = 1; a[1] = 1; a[n_] := a[n] = (4*n*(2*n-1)*(5*n+7)*a[n-1] - 36*(n-1)*(2*n-3)*(2*n-1)*a[n-2]) / ((n+2)*(n+3)^2); Table[a[n], {n, 0, 21}]

N=66; v=vector(N); v[1]=1; v[2]=1;
for(n=2, N-1, my(t=n+1); v[t] = (-36*(n-1)*(2*n-3)*(2*n-1)*v[t-2] + 4*n*(2*n-1)*(5*n+7)*v[t-1]) / ((n+2)*(n+3)^2) );
v \\ Joerg Arndt, Sep 20 2014

a(n) = (-36*(n-1)*(2*n-3)*(2*n-1)*a(n-2) + 4*n*(2*n-1)*(5*n+7)*a(n-1)) / ((n+2)*(n+3)^2).

a(n) = (9*(n+1)*A005802(n)-(n+5)*A005802(n+1))*binomial(2*n,n)/(2*(n+1)*(n+2)). - Mark van Hoeij, Nov 12 2023

cp's OEIS Frontend

A284712 Number of indecomposable permutations avoiding the pattern 4321.

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A129130 Number of triples of standard tableaux with the same shape of height less than or equal to three.

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A145845 Number of permutations of length 2n+1 which are invariant under the reverse-complement map and have no decreasing subsequences of length 5.

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A145846 Number of permutations of length 2n which are invariant under the reverse-complement map and have no decreasing subsequences of length 6.

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A224298 Number of permutations in S_n containing exactly 3 increasing subsequences of length 4.

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A247591 Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.

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