A099390 -id:A099390 - cp's OEIS frontend

A103999 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle.

1, 1, 1, 1, 6, 1, 1, 16, 34, 1, 1, 54, 196, 198, 1, 1, 196, 1666, 2704, 1154, 1, 1, 726, 16384, 64152, 37636, 6726, 1, 1, 2704, 171394, 1844164, 2549186, 524176, 39202, 1, 1, 10086, 1844164, 57523158, 220581904, 101757654, 7300804, 228486, 1

Offset: 0

Ralf Stephan, Feb 26 2005

			Array begins:
  1,   1,     1,        1,           1,             1,                1, ...
  1,   6,    34,      198,        1154,          6726,            39202, ...
  1,  16,   196,     2704,       37636,        524176,          7300804, ...
  1,  54,  1666,    64152,     2549186,     101757654,       4064620168, ...
  1, 196, 16384,  1844164,   220581904,   26743369156,    3252222705664, ...
  1, 726,171394, 57523158, 21050622914, 7902001927776, 2988827208115522, ...

Cliff, Danny and Zoe Stoll, About Klein bottles
W. T. Lu and F. Y. Wu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.

Rows include A003499, A067902+2. Columns include A003500+2.
Main diagonal gives A340557.
Cf. A099390, A103997.

T[m_, n_] := Product[4 Sin[(4k-1) Pi/(4n)]^2 + 4 Cos[j Pi/(2m+1)]^2, {j, 1, m}, {k, 1, n}] // Round;
Table[T[m-n, n], {m, 0, 9}, {n, 0, m}] // Flatten (* Jean-François Alcover, Aug 20 2018 *)

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin((4*a-1)*Pi/(4*n))^2+4*sin((2*b-1)*Pi/(2*k))^2)))} \\ Seiichi Manyama, Jan 11 2021

T(M, N) = Product_{m=1..M} Product_{n=1..N} ( 4sin(Pi*(4n-1)/(4N))^2 + 4sin(Pi*(2m-1)/(2M))^2 ).

A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 15, 36, 15, 13, 1, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 1, 34, 56, 281, 192, 281, 56, 34, 1, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 1, 89, 209, 2245, 2415, 6728, 2415, 2245, 209, 89, 1, 1

Offset: 0

Alois P. Heinz, Apr 15 2011

			A(3,3) = 4, because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
  . .___. . .___. . .___. . .___.
  ._|___| ._|___| ._| | | ._|___|
  | |___| | | | | | |_|_| |___| |
  |_|___| |_|_|_| |_|___| |___|_|
Array begins:
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  2,  3,   5,    8,   13, ...
  1, 1,  3,  4,  11,   15,   41, ...
  1, 1,  5, 11,  36,   95,  281, ...
  1, 1,  8, 15,  95,  192, 1183, ...
  1, 1, 13, 41, 281, 1183, 6728, ...

Alois P. Heinz, Antidiagonals n = 0..75, flattened
Eric Weisstein's World of Mathematics, Perfect Matching
Wikipedia, FKT algorithm
Wikipedia, Matching (graph theory)
Index entries for sequences related to dominoes

Rows m=0+1, 2-12 give: A000012, A000045(n+1), A002530(n+1), A005178(n+1), A189003, A028468, A189004, A028470, A189005, A028472, A210724, A028474.
Main diagonal gives: A189002.
Cf. A099390, A187596, A187616, A187617, A187618, A004003.

with(LinearAlgebra):
A:= proc(m, n) option remember; local i, j, s, t, M;
      if m=0 or n=0 then 1
    elif m1 or j>1 or s=0 then
               if j

A[1, 1] = 1; A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2];M[i_, j_] /; j < i := -M[j, i]; M[, ] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j-s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1-2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m-s, n*m-s}]]]]]; Table[Table[A[m, d-m], {m, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 26 2013, translated from Maple *)

A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 3, 1, 0, 0, 3, 0, 3, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 4, 2, 0, 0, 1, 0, 0, 1, 0, 1, 6, 0, 0, 3, 1, 0, 4, 0, 5, 0, 0, 0, 1, 1, 8, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 6, 4, 0, 0, 1, 0, 6, 1, 0, 6, 5, 0, 6, 3, 1, 2, 10, 0, 0, 1, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. A standard domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(75) = 6 tableaux:
1 2 4   1 2 3   1 2 2   1 1 4   1 1 4   1 1 3
1 2 4   1 2 3   1 3 3   2 3 4   2 2 4   2 2 3
3 3     4 4     4 4     2 3     3 3     4 4

Cf. A000085, A000712, A000898, A001222, A004003, A056239, A099390, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299926, A300060, A300061.

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

Original entry on oeis.org

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 272, 224, 1156, 2, 200, 722, 3108, 1058, 6728, 2, 478, 3108, 9922, 39952, 5054, 39204, 2, 1156, 10082, 90176, 155682, 537636, 24200, 228488, 2, 2786, 39952, 401998, 3113860, 2540032, 7379216, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 18 2021

Dimer tilings of 2n x k toroidal grid.

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     272,      722,       3108, ...
  2,   200,   224,    3108,     9922,      90176, ...
  2,  1156,  1058,   39952,   155682,    3113860, ...
  2,  6728,  5054,  537636,  2540032,  114557000, ...
  2, 39204, 24200, 7379216, 41934482, 4357599552, ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. See Eq. (25).
Index entries for sequences related to dominoes

Columns 1..12 give A007395, A162484(2*n), A231087, A220864(2*n), A231485, A232804(2*n), A230033, A253678(2*n), A281583, A281679(2*n), A308761, A309018(2*n).
Rows 1..6 give A162484, A220864, A232804, A253678, A281679, A309018.
T(n,2*n) gives A335586.
Cf. A341533, A341738, A341739.
Cf. A099390, A103997, A103999.

T(n,k) = A341533(n,k)/2 + A341738(n,k) + 2 * ((k+1) mod 2) * A341739(n,ceiling(k/2)).
T(n, 2k) = T(k, 2n).
If k is odd, T(n,k) = A341533(n,k) = 2*A341738(n,k).

New name from Andrey Zabolotskiy, Dec 26 2021

A028468 Number of perfect matchings in graph P_{6} X P_{n}.

Original entry on oeis.org

1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479

Offset: 0

N. J. A. Sloane, Per H. Lundow

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
R. P. Stanley, Enumerative Combinatorics I, p. 292.

Alois P. Heinz, Table of n, a(n) for n = 0..450
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
David Klarner and Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52.
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 5.
Thotsaporn "Aek" Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
Index entries for linear recurrences with constant coefficients, signature (1,20,10,-38,-10,20,-1,-1).

Row 6 of arrays A099390, A189006.
Column k=2 of A251072.
Cf. A005178.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1)*(x^3+ 6*x^2+5*x+1)) )); // G. C. Greubel, Nov 25 2018

seq(coeff(series((1+2*x-x^2)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(x+1)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 23 2018

a[n_] := Product[2(2 + Cos[(2 k Pi)/7] + Cos[(2 j Pi)/(n+1)]), {k, 1, 3}, {j, 1, n/2}] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 19 2018, after A099390 *)
LinearRecurrence[{1, 20, 10, -38, -10, 20, -1, -1}, {1, 1, 13, 41, 281, 1183, 6728, 31529}, 30] (* Vincenzo Librandi, Nov 24 2018 *)

my(x='x+O('x^30)); Vec(-(x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(1+x)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1))) \\ Altug Alkan, Mar 23 2016

s=((x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 25 2018

From N. J. A. Sloane, Feb 03 2009: (Start)
a(1) = 1,
a(2) = 13,
a(3) = 41,
a(4) = 281,
a(5) = 1183,
a(6) = 6728,
a(7) = 31529,
a(8) = 167089,
a(9) = 817991,
a(10) = 4213133,
a(11) = 21001799,
a(12) = 106912793,
a(13) = 536948224,
a(14) = 2720246633, and
a(n) = 40*a(n-2) - 416*a(n-4) + 1224*a(n-6) - 1224*a(n-8) + 416*a(n-10) - 40*a(n-12) + a(n-14). (From Faase's web page.) (End)
G.f.: (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1) / ( (1-x) *(1+x) *(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1) ).
a(n) = a(n-1)+20*a(n-2)+10*a(n-3)-38*a(n-4)-10*a(n-5)+20*a(n-6)-a(n-7)-a(n-8). - Sergey Perepechko, Sep 23 2018

A028470 Number of perfect matchings in graph P_{8} X P_{n}.

Original entry on oeis.org

1, 1, 34, 153, 2245, 14824, 167089, 1292697, 12988816, 108435745, 1031151241, 8940739824, 82741005829, 731164253833, 6675498237130, 59554200469113, 540061286536921, 4841110033666048, 43752732573098281, 393139145126822985, 3547073578562247994, 31910388243436817641

Offset: 0

Per H. Lundow

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Alois P. Heinz, Table of n, a(n) for n = 0..1048
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
David Klarner, Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 7.
James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2
Index entries for linear recurrences with constant coefficients, signature (1, 76, 69, -921, -584, 4019, 829, -7012, 829, 4019, -584, -921, 69, 76, 1, -1).

Row 8 of array A099390.

a:= n-> (Matrix(16, (i, j)-> `if` (i=j-1, 1, `if` (i=16, [-1, 1, 76, 69, -921, -584, 4019, 829, -7012][min(j, 18-j)], 0)))^n. <>)[10, 1]: seq(a(n), n=0..50);  # Alois P. Heinz, Apr 14 2011

a[n_] := Product[2(2+Cos[(2j Pi)/9] + Cos[(2k Pi)/(n+1)]), {k, 1, n/2}, {j, 1, 4}] // Round; Join[{1}, Array[a, 21]] (* Jean-François Alcover, Aug 11 2018; a(0)=1 prepended by Georg Fischer, Apr 17 2020 *)

{a(n) = sqrtint(polresultant(polchebyshev(8, 2, x/2), polchebyshev(n, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

Recurrence from Faase web site:
a(1) = 1,
a(2) = 34,
a(3) = 153,
a(4) = 2245,
a(5) = 14824,
a(6) = 167089,
a(7) = 1292697,
a(8) = 12988816,
a(9) = 108435745,
a(10) = 1031151241,
a(11) = 8940739824,
a(12) = 82741005829,
a(13) = 731164253833,
a(14) = 6675498237130,
a(15) = 59554200469113,
a(16) = 540061286536921,
a(17) = 4841110033666048,
a(18) = 43752732573098281,
a(19) = 393139145126822985,
a(20) = 3547073578562247994,
a(21) = 31910388243436817641,
a(22) = 287665106926232833093,
a(23) = 2589464895903294456096,
a(24) = 23333526083922816720025,
a(25) = 210103825878043857266833,
a(26) = 1892830605678515060701072,
a(27) = 17046328120997609883612969,
a(28) = 153554399246902845860302369,
a(29) = 1382974514097522648618420280,
a(30) = 12457255314954679645007780869,
a(31) = 112199448394764215277422176953,
a(32) = 1010618564986361239515088848178, and
a(n) = 153a(n-2) - 7480a(n-4) + 151623a(n-6) - 1552087a(n-8) + 8933976a(n-10) - 30536233a(n-12) + 63544113a(n-14) - 81114784a(n-16) + 63544113a(n-18) - 30536233a(n-20) + 8933976a(n-22) - 1552087a(n-24) + 151623a(n-26) - 7480a(n-28) + 153a(n-30) - a(n-32).
G.f.: (1 -43*x^2 -26*x^3 +360*x^4 +110*x^5 -1033*x^6 +1033*x^8 -110*x^9 -360*x^10 +26*x^11 +43*x^12 -x^14) /(1 -x -76*x^2 -69*x^3 +921*x^4 +584*x^5 -4019*x^6 -829*x^7 +7012*x^8 -829*x^9 -4019*x^10 +584*x^11 +921*x^12 -69*x^13 -76*x^14 -x^15 +x^16). - Sergey Perepechko, Nov 22 2012

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

a(0)=1 prepended by Seiichi Manyama, Apr 13 2020

A187618 Triangle T(m,n) read by rows: number of domino tilings of the 2m X 2n grid (0 <= m <= n).

Original entry on oeis.org

1, 1, 2, 1, 5, 36, 1, 13, 281, 6728, 1, 34, 2245, 167089, 12988816, 1, 89, 18061, 4213133, 1031151241, 258584046368, 1, 233, 145601, 106912793, 82741005829, 65743732590821, 53060477521960000, 1, 610, 1174500, 2720246633, 6675498237130, 16848161392724969, 43242613716069407953, 112202208776036178000000

Offset: 0

N. J. A. Sloane, Mar 12 2011

A099390 is the main entry for this problem.

			Triangle begins:
1
1       2
1       5       36
1       13      281     6728
1       34      2245    167089  12988816
1       89      ...

Alois P. Heinz, Rows n = 0..14, flattened
Index entries for sequences related to dominoes

Cf. A099390, A187596, A187616, A187617.

More terms from Nathaniel Johnston, Mar 22 2011

A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

Original entry on oeis.org

1, 1, 21, 269, 4489, 82981, 2995185, 118897973

Offset: 1

Scott R. Shannon, Feb 09 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

			a(1) = 1 as the only way to tile a 1 x 1 square is with a square with dimensions 1 x 1.
a(2) = 1 as the only way to tile a 2 x 2 square is with a square with dimensions 2 x 2.
a(3) = 21. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |   |       |   |       |   |   |           |
  +           +   +---+---+---+   +---+---+   +   +---+---+---+
  |           |   |           |   |       |   |   |           |
  +           +   +           +   +       +   +   +           +
  |           |   |           |   |       |   |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.

Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.

A028471 Number of perfect matchings (or domino tilings) in the graph P_9 X P_2n.

Original entry on oeis.org

1, 55, 6336, 817991, 108435745, 14479521761, 1937528668711, 259423766712000, 34741645659770711, 4652799879944138561, 623139489426439754945, 83456125990631342400791, 11177167872295392172767936, 1496943834332592837945956455, 200483802581126644843760725601

Offset: 0

Per H. Lundow

nonn

Sean A. Irvine, Table of n, a(n) for n = 0..150
J. L. Hock and R. B. McQuistan, A note on the occupational degeneracy for dimers on a saturated two-dimensional lattice space, Discrete Applied Mathematics, 1984, v.8, 101-104.
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Index entries for linear recurrences with constant coefficients, signature (209, -11936, 274208, -3112032, 19456019, -70651107, 152325888, -196664896, 152325888, -70651107, 19456019, -3112032, 274208, -11936, 209, -1).

Cf. A000045, A001835, A005178, A003775, A028468, A028469, A028470.

Row 9 of array A099390.

T[?OddQ, ?OddQ] = 0;
T[m_, n_] := Product[2(2+Cos[2 j Pi/(m+1)]+Cos[2 k Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
a[n_] := T[2n, 9] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 28 2022 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(9, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

a(n) = 209*a(n-1) - 11936*a(n-2) + 274208*a(n-3) - 3112032*a(n-4) + 19456019*a(n-5) - 70651107*a(n-6) + 152325888*a(n-7) - 196664896*a(n-8) + 152325888*a(n-9) - 70651107*a(n-10) + 19456019*a(n-11) - 3112032*a(n-12) + 274208*a(n-13) - 11936*a(n-14) + 209*a(n-15) - a(n-16). - Jay Anderson (horndude77(AT)gmail.com), Apr 07 2007

G.f.: (1 - 154x + 6777x^2 - 123961x^3 + 1132714x^4 - 5684515x^5 + 16401668x^6 - 27757938x^7 + 27757938*x^8 - 16401668x^9 + 5684515x^10 - 1132714x^11 + 123961x^12 -6777x^13 + 154x^14 - x^15)/(1 - 209x + 11936x^2 - 274208x^3 + 3112032x^4 - 19456019x^5 + 70651107x^6 - 152325888x^7 + 196664896x^8 - 152325888x^9 + 70651107x^10 -19456019x^11 + 3112032x^12 - 274208x^13 + 11936x^14 - 209x^15 + x^16). - Sergey Perepechko, Nov 23 2012

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 107, 108, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131

Offset: 1

Gus Wiseman, Mar 12 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is conjectured to be the Heinz numbers of integer partitions in which the odd parts appear as many times in even as in odd positions.

			Sequence of integer partitions whose Young diagram can be tiled by dominos begins: (), (2), (11), (4), (22), (31), (211), (6), (1111), (8), (42), (51), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Wikipedia, Domino tiling

Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
       (-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
       numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[(-1)^Flatten[Position[primeMS[#],_?OddQ]]]===0&] (* Conjectured *)

cp's OEIS Frontend

A103999 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle.

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A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

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A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

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A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

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A028468 Number of perfect matchings in graph P_{6} X P_{n}.

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A028470 Number of perfect matchings in graph P_{8} X P_{n}.

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A187618 Triangle T(m,n) read by rows: number of domino tilings of the 2m X 2n grid (0 <= m <= n).

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A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

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