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

1,5

Comments

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.

Links

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

Formula

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

Example

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.

Maple

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

Mathematica

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

Prog

(PARI) 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

Crossrefs

Columns 1-6 are A000004, A122746(n-2), A305623, A305624, A305625, and A305626.

Row sums are A327091.

Cf. A008277, A019538, A293500, A305621.

Sequence in context: A233670 A089134 A349776 * A269940 A350463 A341339

Adjacent sequences: A305619 A305620 A305621 * A305623 A305624 A305625

Keyword

nonn,tabl,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved