A305621 -id:A305621 - cp's OEIS frontend

A305622 Triangle read by rows: T(n,k) is the number of chiral pairs of rows of n colors with exactly k different colors.

0, 0, 1, 0, 2, 3, 0, 6, 18, 12, 0, 12, 72, 120, 60, 0, 28, 267, 780, 900, 360, 0, 56, 885, 4188, 8400, 7560, 2520, 0, 120, 2880, 20400, 63000, 95760, 70560, 20160, 0, 240, 9000, 93120, 417000, 952560, 1164240, 725760, 181440, 0, 496, 27915, 409140, 2551440, 8217720, 14817600, 15120000, 8164800, 1814400, 0, 992, 85233, 1748220, 14802900, 64614960, 161247240, 239500800, 209563200, 99792000, 19958400

Offset: 1

Robert A. Russell, Jun 06 2018

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

			The triangle begins:
  0;
  0,   1;
  0,   2,     3;
  0,   6,    18,     12;
  0,  12,    72,    120,      60;
  0,  28,   267,    780,     900,     360;
  0,  56,   885,   4188,    8400,    7560,     2520;
  0, 120,  2880,  20400,   63000,   95760,    70560,    20160;
  0, 240,  9000,  93120,  417000,  952560,  1164240,   725760,  181440;
  ...
For T(3,2)=2, the chiral pairs are AAB-BAA and ABB-BBA.  For T(3,3)=3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 1-6 are A000004, A122746(n-2), A305623, A305624, A305625, and A305626.
Row sums are A327091.
Cf. A008277, A019538, A293500, A305621.

with(combinat):
a:=(n,k)->(factorial(k)/2)* (Stirling2(n,k)-Stirling2(ceil(n/2),k)): seq(seq(a(n,k),k=1..n),n=1..11); # Muniru A Asiru, Sep 27 2018

Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

T(n,k) = (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2));
for (n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 27 2018

T(n,k) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
T(n,k) = (A019538(n,k) - A019538(ceiling(n/2),k)) / 2.
T(n,k) = A019538(n,k) - A305621(n,k).
G.f. for column k: k! x^k / (2*Product_{i=1..k}(1-ix)) - k! (x^(2k-1)+x^(2k)) / (2*Product{i=1..k}(1-i x^2)). - Robert A. Russell, Sep 26 2018
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293500(n, i). - Andrew Howroyd, Sep 13 2019

A056310 Number of reversible strings with n beads using exactly three different colors.

Original entry on oeis.org

0, 0, 3, 18, 78, 273, 921, 2916, 9150, 28065, 85773, 259848, 785778, 2367813, 7128201, 21427956, 64382550, 193326105, 580372293, 1741847328, 5227116378, 15684323853, 47059266081, 141189861996

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=3, the three rows are ABC, ACB, and BAC, being respectively equivalent to CBA, BCA, and CAB, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36).

Cf. A032120, A005418.
Column 3 of A305621.
Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.

seq(coeff(series(-3*x^3*(12*x^4-5*x^3-4*x^2+1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)),x,n+1), x, n), n = 1..25); # Muniru A Asiru, Sep 27 2018

k=3; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 3, 18, 78, 273, 921}, 40] (* Vincenzo Librandi, Sep 27 2018 *)

a(n) = A032120(n) - 3*A005418(n+1) + 3.

G.f.: -3*x^3*(12*x^4 - 5*x^3 - 4*x^2 + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)). [Colin Barker, Jul 07 2012]

A056311 Number of reversible strings with n beads using exactly four different colors.

Original entry on oeis.org

0, 0, 0, 12, 120, 780, 4212, 20424, 93360, 409380, 1749780, 7338792, 30394560, 124705140, 508291812, 2061607224, 8332140720, 33585777060, 135116412660, 542785800072, 2178110589600, 8733345234900

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

			For n=4, the 12 rows are 12 permutations of ABCD that do not include any mutual reversals.  Each of the 12 chiral pairs, such as ABCD-DCBA, is then counted just once. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A032121.
Column 4 of A305621.
Equals (A000919 + A056455) / 2 = A000919 - A305624 = A305624 + A056455.

k=4; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)
CoefficientList[Series[12 x^3 (3 x + 1) (8 x^4 - 3 x^3 - 2 x^2 - x + 1) / ((x - 1) (4 x - 1) (3 x - 1) (2 x + 1) (2 x - 1) (3 x^2 - 1) (2 x^2 - 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 26 2018 *)

Equals A032121(n) - 4*A032120(n) + 6*A005418(n+1) - 4.
G.f.: 12*x^4*(3*x+1)*(8*x^4-3*x^3-2*x^2-x+1)/ ((x-1) * (4*x-1) * (3*x-1) * (2*x+1) * (2*x -1) * (3*x^2-1) * (2*x^2-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009 [Corrected by R. J. Mathar, Sep 16 2009]
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A056312 Number of reversible strings with n beads using exactly five different colors.

Original entry on oeis.org

0, 0, 0, 0, 60, 900, 8400, 63000, 417120, 2551560, 14804700, 82764900, 450518460, 2404510500, 12646078200, 65771496000, 339165516120, 1737486149760, 8855359634100, 44952367981500, 227475768907860, 1148269329527100, 5785013373810000, 29100047092479000

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=5, the 60 rows are 60 permutations of ABCDE that do not include any mutual reversals.  Each of the 60 chiral pairs, such as ABCDE-EDCBA, is then counted just once.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-45,-75,695,-575,-3195,5595,4706,-14918,2160,12840,-7200).

Cf. A005418, A032120, A032121, A032122.
Column 5 of A305621.
Equals (A001118 + A056456) / 2 = A001118 - A305625 = A305625 + A056456.

[60*(StirlingSecond(n, 5)+StirlingSecond(Ceiling(n/2), 5)): n in [1..30]]; // Vincenzo Librandi, Sep 30 2018

k=5; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,30}] (* Robert A. Russell, Nov 25 2017 *) adapted
CoefficientList[Series[-60*x^4*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)), {x, 0, 30}], x] (* Stefano Spezia, Sep 29 2018 *)

a(n) = 60*(stirling(n, 5, 2) + stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 27 2018

a(n) = A032122(n) - 5*A032121(n) + 10*A032120(n) - 10*A005418(n+1) + 5.
G.f.: -60*x^5*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)). [Colin Barker, Sep 03 2012]
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=5 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A056313 Number of reversible strings with n beads using exactly six different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64615680, 476515080, 3355679880, 22837101840, 151449674040, 984573656640, 6302070915840, 39847411326600, 249509384858160, 1550188410555960, 9570844671224760

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=6, the 360 rows are 360 permutations of ABCDEF that do not include any mutual reversals.  Each of the 360 chiral pairs, such as ABCDEF-FEDCBA, is then counted just once. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (19, -117, 81, 1883, -5915, -6615, 53235, -30394, -191744, 264852, 258804, -634248, 43920, 505440, -259200).

Cf. A056308, A056322.
Column 6 of A305621.
Equals (A000920 + A056457) / 2 = A000920 - A305626 = A305626 + A056457.

k=6; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)

a(n) = my(k=6); k!/2*(stirling(n, k, 2) + stirling(ceil(n/2), k, 2)); \\ Altug Alkan, Sep 27 2018

a(n) = A056308(n) - 6*A032122(n) + 15*A032121(n) - 20*A032120(n) + 15*A005418(n+1) - 6.
G.f.: 360*x^6*(8*x^2 - x - 1)*(90*x^7 - 9*x^6 - 29*x^5 - 34*x^4 + 15*x^3 + 9*x^2 - x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)*(6*x^2 - 1)). - Colin Barker, Sep 03 2012
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=6 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A326963 Number of length n arrays with entries that cover an initial interval of positive integers counting chiral pairs as equivalent, i.e., the arrays are reversible.

Original entry on oeis.org

1, 1, 2, 8, 39, 277, 2348, 23684, 272955, 3543901, 51124052, 811318628, 14045786139, 263429197837, 5320671508868, 115141595761844, 2657827341263595, 65185383518111581, 1692767331631966292, 46400793659715329348, 1338843898122243225339, 40562412499252848257197

Offset: 0

Andrew Howroyd, Sep 13 2019

nonn

			a(3) = 8 because there are the following arrays up to reversal:
   111,
   112, 121, 122, 212,
   123, 132, 213.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A305621.

a(n) = {sum(k=0, n, k! * (stirling(n, k, 2) + stirling((n+1)\2, k, 2)) / 2)}

a(n) = Sum_{k=0..n} (k!/2) * (Stirling2(n, k) + Stirling2(ceiling(n/2), k)).

A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 6, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 126, 1806, 8400, 16800, 15120, 5040

Offset: 0

Robert A. Russell, Nov 09 2018

Each zero in the data is the beginning of a new row.

Same as A131689, with rows (except for the first) repeated. - Joerg Arndt, Sep 08 2019

			The triangle begins with T(0,0):
1
0 1
0 1
0 1   2
0 1   2
0 1   6     6
0 1   6     6
0 1  14    36     24
0 1  14    36     24
0 1  30   150    240    120
0 1  30   150    240    120
0 1  62   540   1560   1800    720
0 1  62   540   1560   1800    720
0 1 126  1806   8400  16800   15120    5040
0 1 126  1806   8400  16800   15120    5040
0 1 254  5796  40824 126000  191520  141120   40320
0 1 254  5796  40824 126000  191520  141120   40320
0 1 510 18150 186480 834120 1905120 2328480 1451520 362880
For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.

Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.
Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).
Cf. A131689.

Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten

T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.

cp's OEIS Frontend

A305622 Triangle read by rows: T(n,k) is the number of chiral pairs of rows of n colors with exactly k different colors.

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A056310 Number of reversible strings with n beads using exactly three different colors.

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A056311 Number of reversible strings with n beads using exactly four different colors.

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A056312 Number of reversible strings with n beads using exactly five different colors.

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A056313 Number of reversible strings with n beads using exactly six different colors.

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A326963 Number of length n arrays with entries that cover an initial interval of positive integers counting chiral pairs as equivalent, i.e., the arrays are reversible.

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A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

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