cp's OEIS Frontend

In other words, there are no "column rises", where a "column rise" means a pair of adjacent columns where each entry in the left column is strictly less than the adjacent entry in the right column.

This is R(n,k,0) in [Abramson-Promislow].

From Petros Hadjicostas, Sep 09 2019: (Start)

As stated above, in the notation of Abramson and Promislow (1978), we have T(n,k) = R(n, k, t=0).

Let P_k be the set of all lists a = (a_1, a_2, ..., a_k) of integers a_i >= 0, i = 1, ..., k, such that 1*a_1 + 2*a_2 + ... + k*a_k = k; i.e., P_k is the set all integer partitions of k. Then |P_k| = A000041(k).

From Eq. (6), p. 248, in Abramson and Promislow (1978), with t=0, we get T(n,k) = Sum_{a in P_k} (-1)^(k - Sum_{j=1..k} a_j) * (a_1 + a_2 + ... + a_k)!/(a_1! * a_2! * ... * a_k!) * (k! / ((1!)^a_1 * (2!)^a_2 * ... * (k!)^a_k))^n.

The integer partitions of k = 1..10 are listed on pp. 831-832 of Abramowitz and Stegun (1964). We see that, for k = 1..6, the corresponding multinomial coefficients k! / ((1!)^a_1 * (2!)^a_2 * ... * (k!)^a_k) are all distinct; that is, A070289(k) = A000041(k) and A309951(k,s) = A325305(k,s) for s = 0..A000041(k). For 7 <= k <= 10, this is not true anymore; i.e., A070289(k) < A000041(k) for 7 <= k <= 10 (and we conjecture that this is the case for all k >= 7).

From the theory of difference equations, we see that Abramson and Promislow's Eq. (6) on p. 248 (with t=0) implies that Sum_{s = 0..A070289(k)} (-1)^s * A325305(k,s) * T(n-s,k) = 0 for n >= A070289(k) + 1. For k = 1..5, these recurrences give R. H. Hardin's empirical recurrences shown in the Formula section below.

We also have Sum_{s = 0..A000041(k)} (-1)^s * A309951(k,s) * T(n-s,k) = 0 for n >= A000041(k) + 1, but for k >= 7, the recurrence we get (for column k) may not necessarily be minimal.

To derive the recurrence for row n, let y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978). We get 1 + Sum_{k >= 1} T(n,k)*x^k/(k!)^n = 1/f_n(-x), where f_n(x) = Sum_{i >= 0} (x^i/(i!)^n). Matching coefficients, we get Sum_{s = 1..k} T(n,s) * (-1)^(s-1) * binomial(k,s)^n = 1, from which the recurrence in the Formula section follows.

(End)

Column 6 of A212855.

From Petros Hadjicostas, Sep 08 2019: (Start)

Let P_6 be the set of all lists b = (b_1, b_2, b_3, b_4, b_5, b_6) of integers b_i >= 0, i = 1, ..., 6, such that 1*b_1 + 2*b_2 + 3*b_3 + 4*b_4 + 5*b_5 + 6*b_6 = 6; i.e., P_6 is the set all integer partitions of 6. Then |P_6| = A000041(6) = 11.

From Eq. (6), p. 248, in Abramson and Promislow (1978), with t=0, we get a(n) = A212855(n,6) = Sum_{b in P_6} (-1)^(6-Sum_{j=1..6} b_j) * (b_1 + b_2 + b_3 + b_4 + b_5 + b_6)!/(b_1! * b_2! * b_3! * b_4! * b_5! * b_6!) * (6! / ((1!)^b_1 * (2!)^b_2 * (3!)^b_3 * (4!)^b_4 * (5!)^b_5 * (6!)^b_6))^n.

The integer partitions of 6 are listed on p. 831 of Abramowitz and Stegun (1964). We see that the corresponding multinomial coefficients 6! / ((1!)^b_1 * (2!)^b_2 * (3!)^b_3 * (4!)^b_4 * (5!)^b_5 * (6!)^b_6) are all distinct; that is, A070289(6) = A000041(6) = 11 and A309951(6,s) = A325305(6,s) for s = 0..11. (Compare with the comments for A212854.)

Using the information about partitions of 6 in Eq. (6) (with t=0), p. 248, of Abramson and Promislow (1978), we may derive the explicit equation for a(n) shown below.

Using standard results from the theory of difference equations (since the solution is known explicitly), we may derive R. H. Hardin's empirical recurrence. The recurrence is equivalent to Sum_{s = 0..11} (-1)^s * A325305(6,s) * a(n-s) = 0 for n >= 12.

(End)

This array was inspired by R. H. Hardin's recurrences for the columns of array A212855. Rows k=1 to k=5 are due to him, while the remaining rows were computed by Alois P. Heinz.

Row n has length A000041(n) + 1, i.e., one more than the number of partitions of n.

Let R(m,n) := R(m,n,t=0) = A212855(m,n) for m,n >= 1, where R(m,n,t) = LHS of Eq. (6) of Abramson and Promislow (1978, p. 248).

Let P_n be the set of all lists a = (a_1, a_2,..., a_n) of integers a_i >= 0, i = 1,..., n such that 1*a_1 + 2*a_2 + ... + n*a_n = n; i.e., P_n is the set all integer partitions of n. (We use a different notation for partitions than the one in the name of T(n,k).) Then |P_n| = A000041(n) for n >= 0.

We have R(m,n) = A212855(m,n) = Sum_{a in P_n} (-1)^(n - Sum_{j=1..n} a_j) * (a_1 + a_2 + ... + a_n)!/(a_1! * a_2! * ... * a_n!) * (n! / ((1!)^a_1 * (2!)^a_2 * ... * (n!)^a_n))^m.

The recurrence of R. H. Hardin for column n of array A212855 is Sum_{s = 0..|P_n|} (-1)^s * T(n,s) * R(m-s,n) = 0 for n >= 1 and m >= |P_n| + 1.

The above recurrence is correct for all n >= 1, but it is not always a minimal one. For example, it seems to be the minimal one for n = 1,...,6, but not for n = 7 (see A212854). It seems to be minimal whenever every two different partitions of n give different multinomial coefficients.

For n = 7, the partitions (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (0, 2, 1, 0, 0, 0, 0) (i.e., 2 + 2 + 3) and (a_1, a_2, a_3, a_4, a_5, a_6, a_7) = (3, 0, 0, 1, 0, 0, 0) (i.e., 1 + 1 + 1 + 4) give the same multinomial coefficient: 210 = 7!/(2!2!3!) = 7!/(1!1!1!4!). Hence, to find the minimal recurrence for n = 7, we count 210 only once in the set of multinomial coefficients: 1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040. Then the absolute value of the coefficient of a(n-1) in the minimal recurrence is the sum of these multinomial coefficients (i.e., 11271); the absolute value of the coefficient of a(n-2) in the minimal recurrence is the sum of products of every two of them (i.e., 46169368), and so on.

Looking at the multinomial coefficients of the integer partitions of n = 8, 9, 10 on pp. 831-832 of Abramowitz and Stegun (1964), we see that, even in these cases, the above recurrence is not the minimal one. The number of distinct multinomial coefficients among the integer partitions of n is given by A070289.

From Petros Hadjicostas, Sep 08 2019: (Start)

We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=5, n, t=0).

Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.

(End)

From Petros Hadjicostas, Sep 08 2019: (Start)

We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=6, n, t=0).

Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.

(End)

From Petros Hadjicostas, Sep 08 2019: (Start)

We generalize Daniel Suteu's recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=7, n, t=0).

Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.

(End)

A column rise in a matrix M = (m_{i,j}) is a value of j such that m_{i,j} < m_{i,j+1} for all i = 1..n.

From Petros Hadjicostas, Aug 26 2019: (Start)

Let R(m,n) := R(m,n,t=0) = A212855(m,n) for m,n >= 1, where R(m,n,t) = LHS of Eq. (6) of Abramson and Promislow (1978, p. 248).

Let P_n be the set of all lists b = (b_1, b_2,..., b_n) of integers b_i >= 0, i = 1,..., n, such that 1*b_1 + 2*b_2 + ... + n*b_n = n; i.e., P_n is the set all integer partitions of n. Then |P_n| = A000041(n) for n >= 0.

We have a(n) = R(n,n) = A212855(n,n) = Sum_{b in P_n} (-1)^(n - Sum_{j=1..n} b_j) * (b_1 + b_2 + ... + b_n)!/(b_1! * b_2! * ... * b_n!) * (n! / ((1!)^b_1 * (2!)^b_2 * ... * (n!)^b_n)^n.

(End)