A110141 Triangle, read by rows, where row n lists the denominators of unit fraction coefficients of the products of {c_k}, in ascending order by indices of {c_k}, in the coefficient of x^n in exp(Sum_{k>=1} c_k/k*x^k).

1, 1, 2, 2, 6, 2, 3, 24, 4, 3, 8, 4, 120, 12, 6, 8, 4, 6, 5, 720, 48, 18, 16, 8, 6, 5, 48, 8, 18, 6, 5040, 240, 72, 48, 24, 12, 10, 48, 8, 18, 6, 24, 10, 12, 7, 40320, 1440, 360, 192, 96, 36, 30, 96, 16, 36, 12, 24, 10, 12, 7, 384, 32, 36, 12, 15, 32, 8, 362880, 10080, 2160, 960

Offset: 0

Author

Paul D. Hanna, Jul 13 2005

Keywords

• nonn
• tabf
• frac

Comments

Row n starts with n!, after which the following pattern holds. When terms of row n are divided by a list of factorials, with (n-j-1)! repeated A002865(j+1) times in the list as j=1..n-1, the result is the initial terms of A110142. E.g., row 6 is: {720,48,18,16,8,6,5,48,8,18,6}; divide by respective factorials: {6!,4!,3!,2!,2!,1!,1!,0!,0!,0!,0!} with {4!,3!,2!,1!,0!} respectively occurring {1,1,2,2,4} times (A002865), yields the initial terms of A110142: {1,2,3,8,4,6,5,48,8,18,6}.

The term of the sequence corresponding to the product c_1^{n_1}c_2^{n_2}...c_k^{n_k} is equal to the number of elements in the centralizer of a permutation of n_1+2n_2+...+kn_k elements whose cycle type is 1^{n_1}2^{n_2}...k^{n^k}. (This fact is very standard, in particular, for the theory of symmetric functions.) - Vladimir Dotsenko, Apr 19 2009

Multiplying the values of row n by the corresponding values in row n of A102189, one obtains n!. - Jaimal Ichharam, Aug 06 2015

a(n,k) is the number of permutations in S_n that commute with a permutation having cycle type "k". Here, the cycle type of an n-permutation pi is the vector (i_1,...,i_n) where i_j is the number of cycles in pi of length j. These A000041(n) vectors can be ordered in reverse lexicographic order. The k-th cycle type is the k-th vector in this ordering. - Geoffrey Critzer, Jan 18 2019

Examples

			Coefficients [x^n] exp(c1*x + (c2/2)*x^2 + (c3/3)*x^3 + ...) begin:
[x^0]: 1;
[x^1]: 1*c1;
[x^2]: (1/2)*c1^2 + (1/2)*c2;
[x^3]: (1/6)*c1^3 + (1/2)*c1*c2 + (1/3)*c3;
[x^4]: (1/24)*c1^4 + (1/4)*c1^2*c2 + (1/3)*c1*c3 + (1/8)*c2^2 + (1/4)*c4;
[x^5]: (1/120)*c1^5 + (1/12)*c1^3*c2 + (1/6)*c1^2*c3 + (1/8)*c1*c2^2 + (1/4)*c1*c4 + (1/6)*c2*c3 + (1/5)*c5;
[x^6]: (1/720)*c1^6 + (1/48)*c1^4*c2 + (1/18)*c1^3*c3 + (1/16)*c1^2*c2^2 + (1/8)*c1^2*c4 + (1/6)*c1*c2*c3 + (1/5)*c1*c5 + (1/48)*c2^3 + (1/8)*c2*c4 + (1/18)*c3^2 + (1/6)*c6;
forming this triangle of unit fraction coefficients:
1;
1;
2,2;
6,2,3;
24,4,3,8,4;
120,12,6,8,4,6,5;
720,48,18,16,8,6,5,48,8,18,6;
5040,240,72,48,24,12,10,48,8,18,6,24,10,12,7;
40320,1440,360,192,96,36,30,96,16,36,12,24,10,12,7,384,32,36,12,15,32,8;
362880,10080,2160,960,480,144,120,288,48,108,36,48,20,24,14,384,32,36,12,15,32,8,144,40,24,14,162,18,20,9; ...
		

References

• Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford University Press, 1995. [From Vladimir Dotsenko, Apr 19 2009]

Crossrefs

Cf. A000041, A002865, A102189, A110142, A110143 (row sums).

First, second and third entries of each row are given (up to an offset) by A000142, A052849, and A052560 respectively. - Vladimir Dotsenko, Apr 19 2009

Programs

Mathematica

Table[n!/CoefficientRules[n! CycleIndex[SymmetricGroup[n], s]][[All, 2]], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 18 2019 *)

Formula

Number of terms in row n is A000041(n) (partition numbers). The unit fractions of each row sum to unity: Sum_{k=1..A000041(n)} 1/T(n, k) = 1.

a(n,k) = n!/A181897(n,k). - Geoffrey Critzer, Jan 18 2019