Mike Zabrocki - cp's OEIS frontend

A371505 Number of sub-monoids of the monoid of uniform block permutations of size n that contain the symmetric group S_n.

1, 2, 3, 6, 10, 31, 63, 287, 1099, 8640, 62658, 1546891, 29789119, 2525655957

Offset: 1

Mike Zabrocki, Apr 17 2024

Also equal to the number of anti-chains in the poset of integer partitions of k not equal to (1^k) where mu < lambda iff mu is coarser than lambda and r(mu) >= r(lambda) where r(lambda) = smallest part of lambda not equal to 1.

			a(3) = 3 because the uniform block permutations of size 3; S_3; and the monoid consisting of S_3 and the element with one block are the only three sub-monoids.

D. G. FitzGerald, A presentation for the monoid of uniform block permutations, Bulletin of the Australian Mathematical Society, 68(2) (2003), 317--324.
Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, Plethysm and the algebra of uniform block permutations, Alg. Comb., Volume 5, no. 5 (2022), pp. 1165--1203.
Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, The lattice of submonoids of the uniform block permutations containing the symmetric group, arXiv:2405.09710 [math.CO], 2024. See p. 3.

A273396 Indecomposable collections of multisets with a total of n objects having entries {1,2,...,k} for some k<=n or INVERTi transform of A255906.

Original entry on oeis.org

0, 1, 3, 9, 39, 201, 1227, 8305, 61383, 487761, 4131819, 37072361, 350644047, 3482957945, 36220558835, 393329507169, 4450157382383, 52354044069009, 639307054297779, 8090092395577625, 105935581968131399, 1433456549698679385, 20018656224312123051

Offset: 0

Mike Zabrocki, May 21 2016

nonn

A multiset partition of a multiset S is a set of nonempty multisets whose union is S. The total number of multisets of size n and whose entries have all the values in {1,2,...,k} for some k<=n is given by sequence A255906. A multiset partition is decomposable if there exists a value 1<=dd. A multiset partition is called indecomposable otherwise.

			a(3) = 9 because there are 16 multiset partitions, 9 of them are indecomposable ({{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{1,2}}, {{2},{1,2}}, {{1,1,2}}, {{1,2,2}}, {{2},{1,3}}, {{1,2,3}}) and 7 are decomposable ({{1},{1},{2}}, {{1},{2},{2}}, {{1},{2,2}}, {{2},{1,1}}, {{1},{2},{3}}, {{1},{2,3}}, {{3},{1,2}}).

P. A. MacMahon, Combinatory Analysis, vol 1, Cambridge, 1915.

Alois P. Heinz, Table of n, a(n) for n = 0..300
R. Orellana, M. Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672 [math.CO], 2016; or extended abstract, arXiv:1510.00438 [math.CO], 2015.

Cf. A074664, A003319.

INVERTi transform of A255906.

A249565 Number of self-avoiding walks on the truncated square tiling with n steps.

Original entry on oeis.org

1, 3, 6, 12, 22, 42, 80, 152, 284, 536, 988, 1848, 3412, 6352, 11724, 21718, 39952, 73808, 135668, 250188, 459172, 844888, 1548608, 2845186, 5211548, 9563768, 17501272, 32079524, 58660712, 107425356, 196320596, 359232144, 656099656, 1199676412, 2189995764

Offset: 0

Mike Zabrocki, Nov 01 2014

A self-avoiding walk is a sequence of adjacent points in a lattice that are all distinct. The truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. The edge lattice is also referred to as (4,8^2) lattice. It is also the Cayley graph of the Coxeter group generated by three generators {s_0, s_1, s_2} with the relations s_i^2 = 1, s_0 s_2 = s_2 s_0, (s_i s_{i+1})^4 = 1 for i=0,1.

It is conjectured that a(n) is approximately mu^n*n^{11/32} for large n where mu is the connective constant and mu is approximately 1.80883001(6).

			There are 6 paths of length 2 in the truncated square lattice corresponding to the reduced words in the Coxeter group s_0 s_2, s_0 s_1, s_1 s_2, s_1 s_0, s_2 s_0, s_2 s_1.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..47 (from Alm, 2005)
Sven Erick Alm, Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices, J. Phys. A.: Math. Gen., 38 (2005), 2055-2080. Also technical report of the same name, 2004. See Table 2, column (4.8^2).
I. Jensen, and A. J. Guttmann, Self-avoiding walks, neighbour-avoiding walks and trails on semi-regular lattices, J. Phys. A., 31, (1998), 8137-45.
Keh Ying Lin and Chi Chen Chang, Self-avoiding walks on the 4-8 lattice, International Journal of Modern Physics B, 16 (2002), 1241-1246.
Wikipedia, Truncated square tiling
Wikipedia, Connective constant
M. Zabrocki, SAWs and SAPs on the Cayley graph of a group, notes 2014.

Cf. A001411, A001334.

a(20)-a(21) from Mike Zabrocki, Nov 08 2014

a(19)-a(21) corrected based on Alm (2005) and Lin & Chang (2002), more terms added by Andrey Zabolotskiy, Oct 18 2024

A249795 Self-avoiding walks with n steps on the truncated trihexagonal tiling or (4,6,12) lattice.

Original entry on oeis.org

1, 3, 6, 12, 22, 42, 78, 146, 264, 490, 894, 1646, 3012, 5528, 10086, 18476, 33648, 61472, 111702, 203552, 368872, 670538, 1213118, 2201208, 3980380, 7214200, 13044916, 23627064, 42714902, 77316682, 139695536, 252664214, 456138008, 824332804, 1487051098, 2685425808

Offset: 0

Mike Zabrocki, Nov 05 2014

A self-avoiding walk is a sequence of adjacent points in a lattice that are all distinct.
The truncated trihexagonal tiling or (4,6,12) lattice is one of eight semi-regular tilings of the plane. Each vertex of the lattice is adjacent to a square, hexagon and a 12-sided polygon with sides of equal length.
It is also the Cayley graph of the affine G2 Coxeter group generated by three generators {s_0, s_1, s_2} with the relations (s_0 s_2)^2 = (s_0 s_1)^3 = (s_1 s_2)^6 = 1.

			There are 6 paths of length 2 on the (4,6,12) lattice corresponding to the reduced words in the Coxeter group s_0 s_2, s_0 s_1, s_1 s_2, s_1 s_0, s_2 s_0, s_2 s_1.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..47 (from Alm, 2005; terms 0..42 from Sean A. Irvine)
Sven Erick Alm, Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices, J. Phys. A.: Math. Gen., 38 (2005), 2055-2080. Also technical report of the same name, 2004. See Table 2, column (4.6.12).
Sean A. Irvine, Java program (github)
Wikipedia, truncated trihexagonal tiling

Cf. A001411 (square lattice), A001334 (hexagonal lattice), A249565 (truncated square tiling), A326743 (dual, degree 12 vertex), A326744 (dual, degree 6 vertex), A326745 (dual, degree 4 vertex).

a(15)-a(19) corrected by Mike Zabrocki and Sean A. Irvine, Jul 25 2019

More terms from Sean A. Irvine, Jul 25 2019

A129550 Number of real polynomial invariants for the action of 4 copies of U(2) on the fourth tensor power of C^2.

Original entry on oeis.org

1, 1, 8, 20, 98, 293, 1128, 3409, 10846, 30480, 84652, 217677, 544312, 1289225, 2961626, 6528284, 13980717, 28963980, 58464510, 114806429, 220298632, 412950779, 758418342, 1365044296, 2412766496, 4189995629, 7159916414

Offset: 0

Mike Zabrocki, Apr 20 2007

nonn

M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
Nolan R. Wallach, The Hilbert series for measures of entanglement in 4 qubits, Acta Appl. Math. 86(2005),203-220.

Cf. A129548, A129549.

G.f.: (R(q) + q^76*R(1/q))/((1 - q^2)*(1 - q^4)^7*(1 - q^6)^6*(1 - q^8)^4*(1 - q^10)) where R(q) = 1 + 6*q^6 + 46*q^8 + 110*q^10 + 344*q^12 + 844*q^14 + 2154*q^16 + 4606*q^18 + 9397*q^20 + 16848*q^22 + 28747*q^24 + 44580*q^26 + 65366*q^28 + 88036*q^30 + 111909*q^32 + 131368*q^34 + 145676*q^36 + 149860/2*q^38.

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

A129549 Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

Original entry on oeis.org

1, 3, 20, 78, 352, 1365, 5232, 18271, 60598, 187296, 548020, 1515265, 3991204, 10035401, 24210308, 56188768, 125904351, 273044682, 574635828, 1176027747, 2345376048, 4565886531, 8691118644, 16198834634, 29602895824, 53105875363

Offset: 0

Mike Zabrocki, Apr 20 2007

nonn

David Meyer and Nolan Wallach, Invariants for multiple qubits: the case of 3 qubits, Mathematics of quantum computing, Computational Mathematics Series, 77-98, Chapman&Hall/CRC, 2002.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nolan Wallach, The Hilbert series of measures of entanglement for 4 q-bits, Acta Appl. Math. 86(2005), 203-220.

Cf. A000217, A129548.

t1:=1 + 3*q^4 + 20*q^6 + 76*q^8 + 219*q^10 + 654*q^12 +
1539*q^14 + 3119*q^16 + 5660*q^18 + 9157*q^20 +
12876*q^22 + 16177*q^24 + 18275*q^26 +
18275*q^28 + 16177*q^30 + 12876*q^32 +
9157*q^34 + 5660*q^36 + 3119*q^38 + 1539*q^40 +
654*q^42 + 219*q^44 + 76*q^46 + 20*q^48 + 3*q^50 + q^54;
t2:=(1-q^2)^3*(1-q^4)^11*(1-q^6)^6;
t3:=t1/t2;
t4:=subs(q=sqrt(x),t3);
t5:=series(t4,x,30); # N. J. A. Sloane, Jun 17 2011

a(n) = [q^(2n)] (P(q) + q^54*P(1/q))/((1 - q^2)^3*(1 - q^4)^11*(1 - q^6)^6) where P(q) = 1 + 3*q^4 + 20*q^6 + 76*q^8 + 219*q^10 + 654*q^12 + 1539*q^14 + 3119*q^16 + 5660*q^18 + 9157*q^20 + 12876*q^22 + 16177*q^24 + 18275*q^26.

Revised definition from N. J. A. Sloane, Jun 17 2011

A129548 Measures of entanglement in 3-qbits.

Original entry on oeis.org

1, 1, 8, 9, 36, 43, 120, 147, 329, 406, 784, 966, 1680, 2058, 3312, 4026, 6105, 7359, 10648, 12727, 17732, 21021, 28392, 33397, 43953, 51324, 66080, 76636, 96832, 111588, 138720, 158916, 194769, 221901, 268584, 304437, 364420, 411103, 487256, 547239, 642873

Offset: 0

Mike Zabrocki, Apr 20 2007

David Meyer and Nolan Wallach, Invariants for multiple qubits: the case of 3 qubits, Mathematics of quantum computing, Computational Mathematics Series, 77-98, Chapman & Hall/CRC Press, 2002.

Colin Barker, Table of n, a(n) for n = 0..1000
Nolan Wallach, The Hilbert series of measures of entanglement for 4 q-bits, Acta Appl. Math. 86(2005), 203-220.
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Cf. A000217, A129549.

[(2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320 : n in [0..50]]; // Wesley Ivan Hurt, Oct 15 2015

A129548:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320: seq(A129548(n), n=0..50); # Wesley Ivan Hurt, Oct 15 2015

CoefficientList[Series[(x^2 - x + 1)*(x^2 + 1)/((1 - x)^7*(x + 1)^5), {x, 0, 50}], x] (* Wesley Ivan Hurt, Oct 15 2015 *)

Vec(-(x^2-x+1)*(x^2+1)/((x-1)^7*(x+1)^5) + O(x^50)) \\ Colin Barker, Oct 15 2015

a(n) = [x^(2n)] (1+x^4)*(1+x^4+x^8)/((1-x^2)*(1-x^4)^5*(1-x^6)).
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(2*n+11+(-1)^n)*(2*n^3+27*n^2+169*n+387-3*(n^2-5*n-31)*(-1)^n)/184320. - Luce ETIENNE, Oct 15 2015.
From Colin Barker, Oct 15 2015: (Start)
a(n) = (n^6+24*n^5+280*n^4+1920*n^3+7504*n^2+14976*n+11520)/11520 (n even).
a(n) = (n^6+24*n^5+235*n^4+1200*n^3+3319*n^2+4536*n+2205)/11520 (n odd).
G.f.: -(x^2-x+1)*(x^2+1) / ((x-1)^7*(x+1)^5). (End)
a(n) = 2*a(n-1)+4*(n-2)-10*a(n-3)-5*a(n-4)+20*a(n-5)-20*a(n-7)+5*a(n-8)+10*a(n-9)-4*a(n-10)-2*a(n-11)+a(n-12) for n>11. - Wesley Ivan Hurt, Oct 15 2015

A129123 Number of 4-tuples of standard tableau with height less than or equal to 2.

Original entry on oeis.org

1, 1, 2, 17, 98, 882, 7812, 78129, 815474, 8955650, 101869508, 1194964498, 14374530436, 176681194276, 2212121332488, 28145258688369, 363177582488274, 4745064935840178, 62687665026816228, 836447728509168930, 11261240896657686660, 152847558411986548260

Offset: 0

Mike Zabrocki, Mar 29 2007

nonn

Number of pairs of Dyck paths of semilength n with equal midpoint. - Alois P. Heinz, Oct 07 2022

Alois P. Heinz, Table of n, a(n) for n = 0..839
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. A000108, A001405, A003161, A120730, A355481, A357652.

Column k=4 of A357824.

[(&+[((n-2*j+1)/(n-j+1))^4*Binomial(n,j)^4: j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Nov 08 2022

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
a:= n-> add(b(n, n-2*j)^4, j=0..n/2):
seq(a(n), n=0..21);  # Alois P. Heinz, Mar 25 2025

Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^4, {k,0, Floor[n/2]}], {n,0,20}] (* Vaclav Kotesovec, Dec 16 2017 *)

a(n) = sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^4);

from math import comb
def A129123(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**4 for j in range((n>>1)+1)) # Chai Wah Wu, Mar 25 2025

def A129123(n): return sum(((n-2*j+1)/(n-j+1))^4*binomial(n,j)^4 for j in range((n//2)+1))
[A129123(n) for n in range(31)] # G. C. Greubel, Nov 08 2022

a(n) = Sum_{k=0..n} A120730(n,k)^4. - Philippe Deléham, Oct 18 2008
From Vaclav Kotesovec, Dec 16 2017: (Start)
Recurrence: n*(n+1)^3*(15*n^2 - 34*n + 7)*a(n) = 2*n*(90*n^5 - 309*n^4 + 147*n^3 + 124*n^2 - 135*n + 35)*a(n-1) + 4*(n-1)^2*(4*n - 5)*(4*n - 3)*(15*n^2 - 4*n - 12)*a(n-2).
a(n) ~ 3* 2^(4*n - 1/2) / (Pi^(3/2) * n^(7/2)). (End)
a(n) = A357652(n) - A355481(n). - Alois P. Heinz, Oct 13 2022
a(n) = Sum_{j=0..floor(n/2)} ((n-2*j+1)/(n-j+1))^4 * binomial(n,j)^4. - G. C. Greubel, Nov 08 2022
a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^3. - Seiichi Manyama, Mar 25 2025

A124723 Number of ternary Lyndon words with exactly five 1's.

Original entry on oeis.org

2, 12, 56, 224, 806, 2688, 8448, 25344, 73216, 205004, 559104, 1490944, 3899392, 10027008, 25401752, 63504384, 156893184, 383516672, 928514048, 2228433712, 5305794560, 12540968960, 29444014080, 68702699520, 159390262880

Offset: 6

Mike Zabrocki, Nov 05 2006

nonn

			a(7) = 12 because 11111ab, 1111a1b, 111a11b where ab = 22, 23, 32 or 33 are all ternary Lyndon words of length 7 with five 1's.

Cf. A051168, A027376, A124720, A124721, A124722.

a:= n-> (Matrix([[806, 224, 56, 12, 2, 0$5]]). Matrix(10, (i,j)-> `if`(i=j-1, 1, `if`(j=1, [10, -40, 80, -80, 34, -20, 80, -160, 160, -64] [i], 0)))^(n-10))[1,1]: seq(a(n), n=6..30);  # Alois P. Heinz, Aug 04 2008

G.f.: 2*x^6*(1-2*x+3*x^2)*(1-x)^2/(1-2*x^5)/(1-2*x)^5= (1/(1-2*x)^5-1/(1-2*x^5))/5.

cp's OEIS Frontend

User: Mike Zabrocki

A371505 Number of sub-monoids of the monoid of uniform block permutations of size n that contain the symmetric group S_n.

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A273396 Indecomposable collections of multisets with a total of n objects having entries {1,2,...,k} for some k<=n or INVERTi transform of A255906.

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A249565 Number of self-avoiding walks on the truncated square tiling with n steps.

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A249795 Self-avoiding walks with n steps on the truncated trihexagonal tiling or (4,6,12) lattice.

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A129550 Number of real polynomial invariants for the action of 4 copies of U(2) on the fourth tensor power of C^2.

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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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A129549 Dimension of space of measures of entanglement that are homogeneous of degree 2n, for the case of four qubits.

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Maple

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A129548 Measures of entanglement in 3-qbits.

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Magma

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A129123 Number of 4-tuples of standard tableau with height less than or equal to 2.

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