%I #67 Jul 20 2024 17:48:33

%S 1,1,1,3,2,1,1,7,10,9,3,1,1,15,38,53,34,18,4,1,1,31,130,265,261,195,

%T 80,30,5,1,1,63,422,1221,1700,1696,1016,515,155,45,6,1,1,127,1330,

%U 5369,10143,13097,10508,6832,2926,1120,266,63,7,1

%N Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles (1 <= k <= 2n).

%C See A002872 for detailed description.

%C T(m,k) is the number of achiral color patterns in a row or loop of length 2m using exactly k different colors. Two color patterns are equivalent if we permute the colors. - _Robert A. Russell_, Apr 24 2018

%C T(n,k) = coefficient of x^k for A(2,n)(x) in Gilbert and Riordan's article. - _Robert A. Russell_, Jun 14 2018

%H Alois P. Heinz, <a href="/A293181/b293181.txt">Rows n = 1..100, flattened</a> (first 30 rows from Andrew Howroyd)

%H Ira Gessel, <a href="https://mathoverflow.net/q/291720">What is the number of achiral color patterns for a row of n colors containing k different colors?</a>, mathoverflow, Jan 30 2018.

%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.

%F T(n,k) = coefficient of t^k x^n/n! in exp(t*(exp(x)-1)+(1/2)*t^2*(exp(2*x)-1)). - _Ira M. Gessel_, Jan 30 2018

%F T(m,k) = [m>0]*(k*T(m-1,k)+T(m-1,k-1)+T(m-1,k-2)) + [m==0]*[k==0]. - _Robert A. Russell_, Apr 24 2018

%F Conjecture: T(n,k) = R(n,k)-R(n,k-1), with R(n,k) = Sum_{m=0..k} m^n*A000085(m)*A038205(k-m)/(m!*(k-m)!). - _Mikhail Kurkov_, Jun 26 2018

%e Triangle begins:

%e 1, 1;

%e 1, 3, 2, 1;

%e 1, 7, 10, 9, 3, 1;

%e 1, 15, 38, 53, 34, 18, 4, 1;

%e 1, 31, 130, 265, 261, 195, 80, 30, 5, 1;

%e 1, 63, 422, 1221, 1700, 1696, 1016, 515, 155, 45, 6, 1;

%e 1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1;

%e ...

%e For T(2,2)=3, the row patterns are AABB, ABAB, and ABBA. The loop patterns are AAAB, AABB, and ABAB. - _Robert A. Russell_, Apr 24 2018

%t (* Ach[n, k] is the number of achiral color patterns for a row or loop of n

%t colors containing k different colors *)

%t Ach[n_, k_] := Ach[n, k] = Which[0==k, Boole[0==n], 1==k, Boole[n>0],

%t OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],

%t True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]

%t + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]

%t Table[Ach[n, k], {n, 2, 14, 2}, {k, 1, n}] // Flatten

%t (* _Robert A. Russell_, Feb 06 2018 *)

%t Table[Drop[MatrixPower[Table[Switch[j-i, 0, i-1, 1, 1, 2, 1, _, 0],

%t {i, 1, 2n+1}, {j, 1, 2n+1}], n][[1]], 1], {n, 1, 10}] // Flatten

%t (* _Robert A. Russell_, Apr 14 2018 *)

%t Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]

%t + Aeven[m-1, k-2], Boole[m == 0 && k == 0]]

%t Table[Aeven[m, k], {m, 1, 10}, {k, 1, 2m}] // Flatten (* _Robert A. Russell_, Apr 24 2018 *)

%o (PARI) \\ see A056391 for Polya enumeration functions

%o T(n,k) = 2*NonequivalentStructsExactly(CylinderPerms(2,n),k) - stirling(2*n,k,2);

%o (PARI)

%o seq(n)={Vec(serlaplace(exp(y*(exp(x + O(x*x^n))-1)+(1/2)*y^2*(exp(2*x + O(x*x^n))-1))) - 1)}

%o {my(T=seq(10)); for(n=1, #T, for(k=1, 2*n, print1(polcoeff(T[n], k), ", ")); print)} \\ _Andrew Howroyd_, Jan 31 2018

%Y Row sums are A002872.

%Y Maximum row values are A002873.

%Y Number of achiral color patterns of length odd n in A140735.

%Y Column k=3 gives A056182.

%K nonn,tabf

%O 1,4

%A _Andrew Howroyd_, Oct 01 2017