A212851 -id:A212851 - cp's OEIS frontend

A325305 Irregular triangular array, read by rows: T(n,k) is the sum of the products of distinct multinomial coefficients (n_1 + n_2 + n_3 + ...)!/(n_1! * n_2! * n_3! * ...) taken k at a time, where (n_1, n_2, n_3, ...) runs over all integer partitions of n (n >= 0, 0 <= k <= A070289(n)).

1, 1, 1, 1, 1, 3, 2, 1, 10, 27, 18, 1, 47, 718, 4416, 10656, 6912, 1, 246, 20545, 751800, 12911500, 100380000, 304200000, 216000000, 1, 1602, 929171, 260888070, 39883405500, 3492052425000, 177328940580000, 5153150631600000, 82577533320000000, 669410956800000000, 2224399449600000000, 1632586752000000000, 1, 11271

Offset: 0

Petros Hadjicostas, Sep 05 2019

This array was inspired by R. H. Hardin's recurrences for the columns of array A212855. Row n has length A070289(n) + 1.
This array differs from array A309951 starting at row n = 7. Array A309951 calculates a similar sum of products of multinomial coefficients, but the multinomial coefficients do not have to be distinct. Row n of array A309951 has length A000041(n) + 1, i.e., one more than the number of partitions of n.
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.
For n = 1..6, all the multinomial coefficients n!/((1!)^a_1 * (2!)^a_2 * ... * (n!)^a^n) corresponding to lists (a_1,...,a_n) in P_n are distinct; that is, A000041(n) = A070289(n) for n = 1..6.
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, A000041(7) > A070289(7).
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, we have A000041(n) > A070289(n).

			Triangle begins as follows:
  [n=0]: 1,   1;
  [n=1]: 1,   1;
  [n=2]: 1,   3,     2;
  [n=3]: 1,  10,    27,     18;
  [n=4]: 1,  47,   718,   4416,    10656,      6912;
  [n=5]: 1, 246, 20545, 751800, 12911500, 100380000, 304200000, 216000000;
  ...
For example, when n = 3, the integer partitions of 3 are 3, 1+2, 1+1+1, and the corresponding multinomial coefficients are 3!/3! = 1, 3!/(1!2!) = 3, and 3!/(1!1!1!) = 6. They are all distinct. Then T(n=3, k=0) = 1, T(n=3, k=1) = 1 + 3 + 6 = 10, T(n=3, k=2) = 1*3 + 1*6 + 3*6 = 27, and T(n=3, k=3) = 1*3*6 = 18.
Consider the list [1, 7, 21, 35, 42, 105, 140, 210, 420, 630, 840, 1260, 2520, 5040] of the A070289(7) = 15 - 1 = 14 distinct multinomial coefficients corresponding to the 15 integer partitions of 7. Then  T(7,0) = 1, T(7,1) = 11271 (sum of the coefficients), T(7,2) = 46169368 (sum of products of every two different coefficients), T(7,3) = 92088653622 (sum of products of every three different coefficients), and so on. Finally, T(7,14) = 2372695722072874920960000000000 = product of these coefficients.

Alois P. Heinz, Rows n = 0..15, flattened
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6), p. 248.
Wikipedia, Partition (number theory).

Cf. A000012 (column k=0), A000041, A005651, A070289, A212850, A212851, A212852, A212853, A212854, A212855, A212856, A309951, A309972, A325308 (column k=1).

g:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [{map(x->
      binomial(n, i)*x, g(n-i, min(n-i, i)))[], g(n, i-1)[]}[]])
    end:
b:= proc(n, m) option remember; `if`(n=0, 1,
      expand(b(n-1, m)*(g(m$2)[n]*x+1)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(nops(g(n$2)), n)):
seq(T(n), n=0..7);  # Alois P. Heinz, Sep 05 2019

g[n_, i_] := g[n, i] = If[n == 0 || i == 1, {n!}, Union[Map[Function[x, Binomial[n, i] x], g[n - i, Min[n - i, i]]], g[n, i - 1]]];
b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m] (g[m, m][[n]] x + 1)];
T[n_] := CoefficientList[b[Length[g[n, n]], n], x];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, May 06 2020, after Maple *)

Sum_{k=0..A070289(n)} (-1)^k * T(n,k) = 0.

A212806 Number of n X n matrices in which each row is a permutation of [1..n] and which contain no column rises.

Original entry on oeis.org

1, 3, 163, 271375, 21855093751, 128645361626874561, 78785944892341703819175577, 6795588328283070704898044776213094655, 107414633522643325764587104395687638119674465944431, 392471529081605251407320880492124164530148025908765037878553312273, 407934916447631403509359040563002566177814886353044858592046202746464825839911293037

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

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)

			For n=2 the three matrices are [12/21], [21/12], [21/21] (but not [12/12]).
From _Petros Hadjicostas_, Aug 26 2019: (Start)
For example, when n = 3, the integer partitions of 3 are 3, 1+2, and 1+1+1, with corresponding (b_1, b_2, b_3) notation (0,0,1), (1,1,0), and (3,0,0). The corresponding multinomial coefficients are 3!/3! = 1, 3!/(1!*2!) = 3, and 3!/(1!*1!*1!) = 6, while the corresponding quantities (b_1 + b_2 + b_3)!/(b_1!*b_2!*b_3!) are 1, 2, and 1. The corresponding exponents of -1 (i.e., n - Sum_{j=1..n} b_j) are 3 - (0+0+1) = 2, 3 - (1+1+0) = 1, and 3 - (3+0+0) = 0.
It follows that a(n) = (-1)^2 * 1 * 1^3 + (-1)^1 * 2 * 3^3 + (-1)^0 * 1 * 6^3 = 163.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..30 (first 18 terms from R. H. Hardin)
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250. MR0469773 (57 #9554). [Their a(5) on p. 250 is wrong; see A212845.]
Wikipedia, Partition (number theory).

A212805 is a lower bound.
Main diagonal of A212855.
Cf. A000041, A070289, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212858, A212859, A212860, A309951, A325305.

A212806 := proc(n) sum(z^k/k!^n, k=0..infinity);
series(%^x, z=0, n+1): n!^n*coeff(%,z,n); add(abs(coeff(%,x,k)),k=0..n) end:
seq(A212806(n), n=1..11); # Peter Luschny, May 27 2017

a[n_] := Module[{s0, s1, s2}, s0 = Sum[z^k/k!^n, {k, 0, n}]; s1 =  Series[s0^x, {z, 0, n + 1}] // Normal; s2 = n!^n*Coefficient[s1, z, n]; Sum[Abs[Coefficient[s2, x, k]], {k, 0, n}]]; Array[a, 11] (* Jean-François Alcover, Feb 27 2018, after Peter Luschny *)
T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k-j], {j, 1, k}]];
a[n_] := T[n, n];
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)

Abramson and Promislow give a g.f. for R(m,n,t), the number of m X n matrices in which each row is a permutation of [1..n] and which contain exactly t column rises:

1 + Sum_{n>=1} Sum_{t=0..n-1} R(m,n,t) y^t x^n/(n!)^m = (y-1)/(y-f(x(y-1))) where f(x) = Sum_{i>=0} x^i/(i!)^m.

Corrected by R. H. Hardin, May 28 2012

cp's OEIS Frontend

A325305 Irregular triangular array, read by rows: T(n,k) is the sum of the products of distinct multinomial coefficients (n_1 + n_2 + n_3 + ...)!/(n_1! * n_2! * n_3! * ...) taken k at a time, where (n_1, n_2, n_3, ...) runs over all integer partitions of n (n >= 0, 0 <= k <= A070289(n)).

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