A320525 -id:A320525 - cp's OEIS frontend

A320751 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a row of length n using k or fewer colors (subsets).

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 4, 16, 12, 0, 0, 0, 1, 4, 20, 52, 28, 0, 0, 0, 1, 4, 20, 80, 169, 56, 0, 0, 0, 1, 4, 20, 86, 336, 520, 120, 0, 0, 0, 1, 4, 20, 86, 400, 1344, 1600, 240, 0, 0, 0, 1, 4, 20, 86, 409, 1852, 5440, 4840, 496, 0

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
T(n,k)=Xi_k(P_n) which is the number of non-equivalent distinguishing partitions of the path on n vertices, with at most k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. - Bahman Ahmadi, Sep 02 2019

			Array begins with T(1,1):
0   0     0      0       0       0       0       0       0       0 ...
0   0     0      0       0       0       0       0       0       0 ...
0   1     1      1       1       1       1       1       1       1 ...
0   2     4      4       4       4       4       4       4       4 ...
0   6    16     20      20      20      20      20      20      20 ...
0  12    52     80      86      86      86      86      86      86 ...
0  28   169    336     400     409     409     409     409     409 ...
0  56   520   1344    1852    1976    1988    1988    1988    1988 ...
0 120  1600   5440    8868   10168   10388   10404   10404   10404 ...
0 240  4840  21760   42892   54208   57108   57468   57488   57488 ...
0 496 14641  87296  210346  299859  331705  337595  338155  338180 ...
0 992 44044 349184 1038034 1699012 2012202 2091458 2102518 2103348 ...
For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA.
For T(4,3)=4, the above, AABC-ABCC, and ABAC-ABCB.

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Columns 1-6 are A000004, A122746(n-3), A107767(n-1), A320934, A320935, A320936.
As k increases, columns converge to A320937.
Cf. transpose of A278984 (oriented), A320750 (unoriented), A305749 (achiral).
Partial column sums of A320525.

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
Table[Sum[StirlingS2[n,j] - Ach[n,j], {j,k-n+1}]/2, {k,15}, {n,k}] // Flatten

T(n,k) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where S2 is the Stirling subset number A008277 and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
T(n,k) = (A278984(k,n) - A305749(n,k)) / 2 = A278984(k,n) - A320750(n,k) = A320750(n,k) - A305749(n,k).
T(n,k) = Sum_{j=1..k} A320525(n,j).

A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).

Original entry on oeis.org

0, 0, 0, 2, 10, 40, 141, 464, 1480, 4600, 14145, 43052, 130480, 393820, 1186521, 3568784, 10725760, 32213200, 96714465, 290284052, 871142800, 2613981700, 7843080201, 23531425304, 70598731840, 211804847800, 635432109585, 1906330676252, 5719061512720, 17157321139180

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(4)=2, the two chiral pairs are AABC-ABCC and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (6, -6, -24, 49, 6, -66, 36).

Column 3 of A320525.

Cf. A000392 (oriented), A056327 (unoriented), A304973 (achiral).

I:=[0,0,0,2,10,40,141]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

k=3; Table[(StirlingS2[n,k] - If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n, 1, 30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 3; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 0, 2, 10, 40,
  141}, 40]

m=40; v=concat([0,0,0,2,10,40,141], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=3 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^3 / Product_{k=1..3} (1 - k*x) - x^3*(1 + 2 x)/((1 - 2 x^2)*(1 - 3 x^2))) / 2.
a(n) = (A000392(n) - A304973(n)) / 2 = A000392(n) - A056327(n) = A056327(n) - A304973(n).

A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 4, 28, 167, 824, 3840, 16920, 72655, 305140, 1265264, 5193188, 21173607, 85887984, 347150080, 1399355440, 5629755935, 22615859180, 90754215024, 363888497148, 1458169977847, 5840524999144, 23385639542720, 93613165023560, 374664497695215, 1499293455643620, 5999080285068784, 24002040333605908

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.

Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).

Col. 4 of A320525.

Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral).

k=4; Table[(StirlingS2[n,k] - If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/2, {n,30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 4; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 0, 4, 28, 167}, 30]

a(n) = (S2(n,k) - A(n,k))/2, where k=4 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^4 / Product_{k=1..4} (1 - k*x) - x^4*(1 + x)^2*(1 - 2 x^2) / Product_{k=1..4} (1 - k*x^2)) / 2.
a(n) = (A000453(n) - A304974(n)) / 2 = A000453(n) - A056328(n) = A056328(n) - A304974(n).

A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).

Col. 5 of A320525.

Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).

I:=[0,0,0,0,0,6,64,508,3428,21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018

k=5; Table[(StirlingS2[n,k] - If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]

m=30; v=concat([0,0,0,0,0,6,64,508,3428,21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.
a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).
a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020

A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 1, 12, 17, 4, 0, 0, 0, 2, 43, 82, 49, 9, 0, 0, 0, 7, 137, 388, 339, 125, 15, 0, 0, 0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0, 0, 31, 1190, 6405, 10900, 7928, 2761, 490, 35, 0, 0, 0, 57, 3394, 24853, 56586, 54272, 25609, 6365, 850, 51, 0, 0

Offset: 1

Bahman Ahmadi, Sep 04 2019

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with exactly k parts.
Number of n-bead bracelet structures using exactly k different colored beads that are not self-equivalent under either a nonzero rotation or reversal (turning over bracelet). Comparable sequences for unoriented (reversible) strings and necklaces (cyclic group) are A320525 and A327693. - Andrew Howroyd, Sep 23 2019

			Triangle begins:
  0;
  0,  0;
  0,  0,   0;
  0,  0,   0,    0;
  0,  0,   0,    0,    0;
  0,  0,   4,    2,    0,    0;
  0,  1,  12,   17,    4,    0,   0;
  0,  2,  43,   82,   49,    9,   0,  0;
  0,  7, 137,  388,  339,  125,  15,  0, 0;
  0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0;
  ...
For n=7, we can partition the vertices of the cycle C_7 with exactly 3 parts, in 12 ways, such that all these partitions are distinguishing for C_7 and that all the 12 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2, 3 }, { 4, 5, 6, 7 } },
    { { 1 }, { 2, 3, 4, 6 }, { 5, 7 } },
    { { 1 }, { 2, 3, 4, 7 }, { 5, 6 } },
    { { 1 }, { 2, 3, 5, 6 }, { 4, 7 } },
    { { 1 }, { 2, 3, 5, 7 }, { 4, 6 } },
    { { 1 }, { 2, 3, 6 }, { 4, 5, 7 } },
    { { 1 }, { 2, 3, 7 }, { 4, 5, 6 } },
    { { 1 }, { 2, 4, 5, 6 }, { 3, 7 } },
    { { 1 }, { 2, 4, 7 }, { 3, 5, 6 } },
    { { 1, 2 }, { 3, 4, 6 }, { 5, 7 } },
    { { 1, 2 }, { 3, 5, 6 }, { 4, 7 } },
    { { 1, 2, 4 }, { 3, 6 }, { 5, 7 } }.
From _Andrew Howroyd_, Sep 23 2019: (Start)
For n=6, k=4 the partitions are:
    { { 1, 2, 4 }, { 3 }, { 5 }, { 6 } },
    { { 1, 2 }, { 3, 5 }, { 4 }, { 6 } }.
These correspond to the bracelet structures AABACD and AABCBD.
(End)

Bahman Ahmadi, GAP Program
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Column k=2 is A327734.
Row sums are A327740.
Cf. A309784, A309785, A320525, A320748, A152176, A320647, A324803, A327693.

T(n,k) = A324803(n,k) - A324803(n,k-1).

a(56)-a(78) from Andrew Howroyd, Sep 23 2019

A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638, 210332227, 1367416232, 8752773288, 55343303064, 346540112781, 2153037307846, 13292835205606, 81652655795106, 499484899831775, 3045117929546220, 18513208314957356, 112297592929814292, 679900657841661529, 4110073054119135194, 24814158520762637754

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.

Index entries for linear recurrences with constant coefficients, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).

Column 6 of A320525.

Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).

m:=30; R:=PowerSeriesRing(Integers(), m); [0,0,0,0,0,0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // G. C. Greubel, Oct 19 2018

k=6; Table[(StirlingS2[n,k] - If[EvenQ[n], StirlingS2[n/2+3,6] - 3StirlingS2[n/2+2,6] - 8StirlingS2[n/2+1,6] + 16StirlingS2[n/2,6], 3StirlingS2[(n+5)/2,6] - 17StirlingS2[(n+3)/2,6] + 20StirlingS2[(n+1)/2,6]])/2, {n,30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 6; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]

x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1,6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1,6,(1-k*x^2)))/2)) \\ G. C. Greubel, Oct 19 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=6 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^6 / (Product_{k=1..6} (1 - k*x)) - x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).

A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1988, 10404, 57488, 338180, 2103378, 13814202, 95423766, 691415451, 5239857008, 41431883216, 341036489096, 2916365967707, 25862060748614, 237434856965694, 2253357681164288, 22076002386446896, 222979432604192844, 2319295160051570620

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of triangle A320525.
Limit as k increases of column k of array A320751.
Cf. A000110 (oriented), A103293 (unoriented), A080107 (achiral).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,n}]/2,{n,40}]

\\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n,n,i,k,i>=k)); for(i=3, n, for(k=2, n, M[i,k]=k*M[i-2,k] + M[i-2,k-1] + if(k>2, M[i-2,k-2]))); M}
seq(n)={my(A=Ach(n)); vector(n, n, sum(k=1, n, stirling(n,k,2) - A[n,k])/2)} \\ Andrew Howroyd, Sep 18 2019

a(n) = (A000110(n) + A080107(n)) / 2 = A000110(n) - A103293(n+1) = A103293(n+1) - A080107(n).

a(n) = Sum_{j=1..n} (S2(n,j) - Ach(n,j)) / 2, where S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).

cp's OEIS Frontend

A320751 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a row of length n using k or fewer colors (subsets).

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A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).

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A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

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A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

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A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

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A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

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A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

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