A342245 -id:A342245 - cp's OEIS frontend

A381930 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1,2} such that I(x) + W_0(x)W_1(x) + W_0(x)W_2(x) + W_1(x)*W_2(x) = k where I(x) is the number of inversions in x and W_i(x) is the number of occurrences of the letter i in x for i={0,1,2}, n>=0, 0<=k<=floor(2n^2/3).

1, 3, 3, 3, 3, 3, 0, 6, 7, 8, 2, 1, 3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3, 3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3, 3, 0, 0, 0, 0, 6, 6, 6, 12, 18, 27, 33, 52, 62, 77, 82, 86, 75, 68, 48, 35, 19, 11, 2, 1

Offset: 0

Geoffrey Critzer, Mar 10 2025

Sum_{k>=0} T(n,k)*2^k = A342245(n).

Sum_{k>=0} T(n,k)*q^k = the number of ordered pairs (S,T) of idempotent n X n matrices over GF(q) such that ST=TS=S.

			Triangle T(n,k) begins:
  1;
  3;
  3, 3, 3;
  3, 0, 6, 7, 8,  2,  1;
  3, 0, 0, 6, 9, 12, 18, 12, 12,  6,  3;
  3, 0, 0, 0, 6,  6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3;
  ...
T(3,3) = 7 because we have: {0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {1, 0, 1}, {1, 2, 1}, {2, 0, 2}, {2, 1, 2}.

Alois P. Heinz, Rows n = 0..45, flattened

Cf. A000244 (row sums), A027472, A056449, A129529, A342245, A381899.

b:= proc(n, i, j, k) option remember; expand(
     `if`(n=0, z^(i*j+i*k+j*k), b(n-1, i+1, j, k)*z^(j+k)+
      b(n-1, i, j+1, k)*z^k +b(n-1, i, j, k+1)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 10 2025

nn = 6; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, q] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^3, {z, 0, nn}],z]] // Grid

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^3 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).
From Alois P. Heinz, Mar 10 2025: (Start)
Sum_{k>=0} k * T(n,k) = 9 * A027472(n+1).
Sum_{k>=0} (-1)^k * T(n,k) = A056449(n). (End)

A357410 a(n) is the number of covering relations in the poset P of n X n idempotent matrices over GF(2) ordered by A <= B if and only if AB = BA = A.

Original entry on oeis.org

0, 1, 12, 224, 6960, 397792, 42001344, 8547291008, 3336917303040, 2565880599084544, 3852698988517260288, 11517943538435677485056, 67829192662051610706309120, 799669932659456441970547744768, 18652191511341505602408972738871296, 873360272626100960024734923878091948032

Offset: 0

Geoffrey Critzer, Sep 26 2022

nonn

The order given for P in the title is equivalent to the ordering: A <= B if and only if image(A) is contained in image(B) and null(A) contains null(B). Then A is covered by B if and only if there is a 1-dimensional subspace U such that image(B) = image(A) direct sum U. If such a subspace U exists then it is unique and is equal to the intersection of null(A) with image(B). The number of maximal chains in P is A002884(n). The set of all idempotent matrices over GF(q) with this ordering is a binomial poset with factorial function |GL_n(F_q)|/(q-1)^n. (see Stanley reference).

			a(2) = 12 because there are A132186(2) = 8 idempotent 2 X 2 matrices over GF(2).  The identity matrix covers 6 rank 1 matrices each of which covers the zero matrix for a total of 12 covering relations.  Cf. A296548.

R. P. Stanley, Enumerative Combinatorics, Vol I, second edition, page 320.

Wikipedia, Covering relation

Cf. A132186, A342245, A002884, A296548.

nn = 15; B[q_, n_] := Product[q^n - q^i, {i, 0, n - 1}]/(q - 1)^n;
e[q_, u_] := Sum[u^n/B[q, n], {n, 0, nn}];Table[B[2, n], {n, 0, nn}] CoefficientList[Series[e[2, u] u e[2, u], {u, 0, nn}], u]

Sum_{n>=0} a(n) x^n/B(n) = x * (Sum_{n>=0} x^n/B(n))^2 where B(n) = A002884(n).

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