cp's OEIS Frontend

The triangle consists of selected entries from A181897.

Each permutation in the symmetric group S_N has a cycle type specified by an integer partition of N. We encode a partition of N as an N-tuple of multiplicities of j=1..N; e.g., the partition 8 = 1+1+2+4 is encoded as (2,1,0,1,0,0,0,0), abbreviated (2,1,0,1). In general (m_j: j=1..N), and this partition corresponds to a cycle type (a)(b)(cd)(efgh) in S_8. Derangements have no fixed points, hence correspond to partitions with no addends of 1; i.e., m_1=0. A181897 counts all permutations via cycle type (not just derangements), with the partitions ordered reverse lexicographically in terms of their multiplicities N-tuples. E.g., partitions of 4 are ordered: (4), (2,1), (1,0,1), (0,0,0,1). In fact, the columns of A181897 (as a triangular array) are scaled columns of binomial coefficients binomial(p,q) with q fixed for each column, scaled by the entries in this listing (see new comment in A181897 for details).

The number of terms in the n-th row of this irregular triangle is given by p(n+1)-p(n) where p(n) = A000041(n) is the partitions counting function. The row sum of row n is A000166(n+1). The number of 'parts' of a partition P is K(P) := Sum_{j=1..N} m_j; if this sequence is signed by (-1)^(1+N+K), then the signed sum of row n is equal to n. The last entry in row n is n!. The first entry in odd rows n is equal to n!!. The first entry in even rows n is equal to n*(n+1)!!/3.

Agrees largely with A101029. The first difference is at n=8 for which a(8)/A101029(8) = 5. There are also other differences; the ratio of the two series entries appears to be always an integer.

For N+1 = i+j+k, let P(N+1;i,j,k) = (N+1-i)*P(N;i-1,j,k) + (N+1-j)*P(N;i,j-1,k) + (N+1-k)*P(N;i,j,k-1), with P(N;i,j,k) invariant upon permutation of the indices i,j,k, also P(N;N,0,0)=1 and P(N;i,j,k) = 0 if i or j or k is negative. The indexing of these values is shown explicitly in the examples.

The row sums are the second-order Eulerian numbers, A008517; precisely, Sum_{(j,k)|j+k=N-i} P(N;i,j,k) = <> = T(N+1,i+1) of A008517. The row sum of row i=N of slice N is (N+1)!. The sum of all entries in slice N is (2*N+1)!!. The edges of the N-th triangular slice of the pyramid are row (N+1) of the first-order Eulerian triangle, A008292.

Number the rows of the triangle beginning with n=0. For each row construct a degree n polynomial with regularly decreasing powers, denoting the polynomial as f_n(x); e.g., for row 2 we have f_2(x)=1x^2+2x+0. Then construct g_n(x)=x^2*f_{n-1}(x)-(n+1)x+1. It obtains that g_n(x)=(1-x)(2-(1+x)^n). These g_n(x) are the denominators of the generating functions for the following sequences: A024537 (n=2); A195350 (n=3); A301417 (n=4); A301420 (n=5); A301421 (n=6); A301424 (n=7). For these sequences the asymptotic term-to-term ratios are 1/(2^(1/n)-1). The numerators of the generating functions are 1-x(x+1)^(n-1).

a(n+1)/a(n) approaches 1/(2^(1/3)-1).

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 5 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/5)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 6 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/6)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 7 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/7)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) The sums of the negative coefficients are 1 less than the corresponding sums of the positive coefficients. See extended comment in A301417.

Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 4 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/4)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.)

More precisely, given a finite collection X:=(x(i), i =1..n) of data, the Girard-Waring formulae express the sum of the k-th powers of the data, S_k(X):=Sum(x(i)^k, i=1..n), in terms of the elementary symmetric polynomials in the data. The j-th elementary symmetric polynomial is s_j(X):=Sum(Product(x(i), x(i) in X_0), X_0 \subseteq X, where |X_0|=j). So the Girard-Waring formulae provide coefficients a(J,k) such that S_k(X)=Sum(a(J,k)*Product(s_j(X), j \in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). [Thus J is an integer partition of k.] By "mean powers" I mean T_k(X):=Sum(x(i)^k, i=1..n)/n. By the "mean symmetric polynomials" I mean t_j(X):=s_j(X)/binomial(n,j). The Girard-Waring mean formulae then provide coefficients b(J,k,n) such that T_k(X)=Sum(b(J,k,n)*Product(t_j(X), j in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). So the sums of positive coefficients that I reference, for a fixed data set size n, and a fixed power k, are Sum(b(J,k,n), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k, such that b(J,k,n)>0).