Geoffrey Critzer - cp's OEIS frontend

A385355 Triangular array read by rows: T(n,k) is the number of n X n matrices A over GF(2) such that the dimension of the null space of A^n is equal to k, n>=0, 0<=k<=n.

1, 1, 1, 6, 6, 4, 168, 168, 112, 64, 20160, 20160, 13440, 7680, 4096, 9999360, 9999360, 6666240, 3809280, 2031616, 1048576, 20158709760, 20158709760, 13439139840, 7679508480, 4095737856, 2113929216, 1073741824, 163849992929280, 163849992929280, 109233328619520, 62419044925440, 33290157293568, 17182016667648, 8727373545472, 4398046511104

Offset: 0

Geoffrey Critzer, Jun 26 2025

Conjecture: In the limit as n goes to infinity the probability that the nullity of such a random matrix is equal to k is Product_{i>=1} (1-1/2^i * 2^binomial(k,2)/A005329(k)).

			Triangle T(n,k) begins:
        1;
        1,       1;
        6,       6,       4;
      168,     168,     112,      64;
    20160,   20160,   13440,    7680,    4096;
  9999360, 9999360, 6666240, 3809280, 2031616, 1048576;
  ...

Cf. A002884 (column k=0), A053763 (main diagonal), A002416 (row sums), A005329.

nn = 6; q = 2; b[p_, i_] := Count[p, i]; d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] := Product[Product[ q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; l= Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; \[Gamma][n_,q_] := Product[q^n - q^i, {i, 0, n - 1}];g[u_, v_, deg_, partitions_] := Total[Map[v^Total[#] u^(deg Total[#])/aut[deg, #] &, partitions]];Map[Select[#, # > 0 &] &,Table[\[Gamma][n, q], {n, 0, nn}] CoefficientList[Series[g[u, v, 1, l]*g[u, 1, 1, l] Product[g[u, 1, deg, l]^\[Nu][[deg]], {deg, 2, nn}], {u, 0, nn}], {u,v}]] // Grid

A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

Original entry on oeis.org

1, 3, 210, 234360, 4047865920, 1092146608143360, 4650098142288472473600, 314462403262051153026062745600, 338960040818652280796119613717033779200, 5834618256563872511581456247120956565738854809600, 1605370810586153268821245248112723240374305354675084328960000

Offset: 0

Geoffrey Critzer, May 17 2025

nonn

Let M be a 2n X 2n matrix over GF(2) such that the column space of M is equal to the null space of M. Then M is idempotent and nullity(M) = n and index(M) = 2. If M' is similar to M then the column space of M' equals the null space of M'. Moreover, all such matrices are in the same similarity class (see Hoffman link).

			a(1) = 3 because there are 3 matrices of size 2 X 2 over GF(2) with the desired property: {{0, 0}, {1, 0}}, {{0, 1}, {0, 0}}, {{1, 1}, {1, 1}}.

D. G. Hoffman, Digraphs of finite linear transformations, Australasion Journal of Combinatorics,12: 225-238 (1995).

Cf. A002884, A006098, A346214, A053763.

q = 2; b[p_, i_] := Count[p, i]; d[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; Table[Product[2^(2 k) - 2^i, {i, 0, (2 k) - 1}]/aut[1, Table[2, {k}]], {k,0, 10}]

a(n) = A002884(n)*A006098(n).

A383656 Triangular array read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k connected components, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 8, 11, 6, 1, 0, 52, 60, 35, 10, 1, 0, 502, 472, 255, 85, 15, 1, 0, 6824, 5166, 2422, 805, 175, 21, 1, 0, 127166, 76712, 30072, 9177, 2100, 322, 28, 1, 0, 3205924, 1526910, 486800, 129360, 28497, 4788, 546, 36, 1, 0, 108975934, 40603534, 10292970, 2285240, 455805, 76629, 9870, 870, 45, 1

Offset: 0

Geoffrey Critzer, May 04 2025

For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).

			Triangle begins:
 1;
 0, 1;
 0, 1, 1;
 0, 2, 3, 1;
 0, 8, 11, 6, 1;
 0, 52, 60, 35, 10, 1;
 0, 502, 472, 255, 85, 15, 1;
 ...

David Bevan, Gi-Sang Cheon, and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission].

Cf. A323843 (column k=1), A135922 (row sums).

nn = 8; Prepend[Table[(Range[0, nn]! CoefficientList[Series[(Exp[-x] g[x])^y, {x, 0, nn}], {x, y}])[[i,1 ;; i]], {i, 2, nn + 1}], {1}] // Grid

E.g.f.: f(x)^y where f(x) is the e.g.f. for A135922.

A383655 Triangle read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k isolated points, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 11, 8, 6, 0, 1, 72, 55, 20, 10, 0, 1, 677, 432, 165, 40, 15, 0, 1, 8686, 4739, 1512, 385, 70, 21, 0, 1, 152191, 69488, 18956, 4032, 770, 112, 28, 0, 1, 3632916, 1369719, 312696, 56868, 9072, 1386, 168, 36, 0, 1, 118317913, 36329160, 6848595, 1042320, 142170, 18144, 2310, 240, 45, 0, 1

Offset: 0

Geoffrey Critzer, May 04 2025

For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).

			Triangle T(n,k) begins:
   1;
   0,  1;
   1,  0,  1;
   2,  3,  0,  1;
  11,  8,  6,  0, 1;
  72, 55, 20, 10, 0, 1;
  ...

David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission].

Cf. A323842 (column k=0), A135922 (row sums).

nn = 10; g[x_] :=Total[Table[Sum[QBinomial[n, k, 2] x^n/n!, {k, 0, n}], {n, 0, nn}]]; Table[(Range[0, nn]! CoefficientList[Series[Exp[y x] Exp[-x] g[x] Exp[-x], {x, 0, nn}], {x, y}])[[i, 1 ;; i]], {i, 1, nn + 1}] // Grid

E.g.f.: exp((y-1)*x)*f(x) where f(x) is the e.g.f. for A135922.

A382223 Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u=0, k>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 16, 12, 4, 1, 0, 1, 67, 66, 22, 5, 1, 0, 1, 374, 513, 172, 35, 6, 1, 0, 1, 2825, 5769, 1969, 355, 51, 7, 1, 0, 1, 29212, 95706, 33856, 5380, 636, 70, 8, 1, 0, 1, 417199, 2379348, 893188, 125090, 12006, 1036, 92, 9, 1

Offset: 0

Geoffrey Critzer, Mar 23 2025

			 1, 1,   1,    1,     1,      1,      1,...
 0, 1,   2,    3,     4,      5,      6,...
 0, 1,   5,   12,    22,     35,     51,...
 0, 1,  16,   66,   172,    355,    636,...
 0, 1,  67,  513,  1969,   5380,  12006,...
 0, 1, 374, 5769, 33856, 125090, 352476,...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; arXiv preprint, arXiv:1909.01550 [math.CO], 2019-2020. See Table 2.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.

Cf. A006116 column k=2, A289539 column k=3, A005329, A382363.

nn = 6; B[n_] := QFactorial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; zetapolys = Drop[Map[Expand[InterpolatingPolynomial[#, x]] &,Transpose[Table[Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^k, {z, 0, nn}], z], {k, 1, nn}]]], -1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid

Sum_{n>=0} T(n,k)/A005329(n) = e(x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).

A382363 Rectangular array read by antidiagonals, T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] such that for all u,v in [n], u->v implies u<=v and c(u)<=c(v), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 8, 7, 3, 1, 0, 64, 44, 15, 4, 1, 0, 1024, 508, 129, 26, 5, 1, 0, 32768, 10976, 1962, 284, 40, 6, 1, 0, 2097152, 450496, 54036, 5371, 530, 57, 7, 1, 0, 268435456, 35535872, 2747880, 180424, 11995, 888, 77, 8, 1, 0, 68719476736, 5435551744, 262091808, 10997576, 476165, 23409, 1379, 100, 9, 1

Offset: 0

Geoffrey Critzer, Mar 23 2025

			  1,    1,     1,     1,      1,      1,       1,...
  0,    1,     2,     3,      4,      5,       6,...
  0,    2,     7,    15,     26,     40,      57,...
  0,    8,    44,   129,    284,    530,     888,...
  0,   64,   508,  1962,   5371,  11995,   23409,...
  0, 1024, 10976, 54036, 180424, 476165, 1072854,...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; arXiv preprint, arXiv:1909.01550 [math.CO], 2019-2020; See Table 2.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.

Cf. A382223, A006125 (column k=1).

nn = 6; B[n_] := QFactorial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; zetapolys = Drop[Map[Expand[InterpolatingPolynomial[#, x]] &,Transpose[Table[Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-z]^k, {z, 0, nn}], z], {k,1,nn}]]],-1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid

Sum_{n>=0} T(n,k)/A005329(n) = 1/e(-x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).

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).

Original entry on oeis.org

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)

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 0, 2, 2, 2, 2, 0, 0, 2, 3, 3, 4, 1, 1, 2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 6, 9, 7, 7, 5, 4, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 4, 4, 8, 6, 10, 12, 14, 12, 14, 10, 10, 6, 4, 2, 2

Offset: 0

Geoffrey Critzer, Mar 09 2025

Sum_{k>=0} T(n,k)*2^k = A132186(n).
Sum_{k>=0} T(n,k)*3^k = A053846(n).
Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
From Alois P. Heinz, Mar 09 2025: (Start)
Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)

			Triangle T(n,k) begins:
  1;
  2;
  2, 1, 1;
  2, 0, 2, 2, 2;
  2, 0, 0, 2, 3, 3, 4, 1, 1;
  2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
  ...
T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.

Alois P. Heinz, Rows n = 0..50, flattened

Row sums give A000079.

Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.

b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
     `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
         expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 09 2025

nn = 7; 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]^2, {z, 0, nn}],z]]

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 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).

A381299 Irregular triangular array read by rows. T(n,k) is the number of ordered set partitions of [n] with exactly k descents, n>=0, 0<=k<=binomial(n,2).

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 4, 1, 8, 12, 18, 18, 12, 6, 1, 16, 32, 60, 84, 100, 92, 76, 48, 24, 8, 1, 32, 80, 176, 300, 448, 572, 650, 658, 596, 478, 334, 206, 102, 40, 10, 1, 64, 192, 480, 944, 1632, 2476, 3428, 4300, 5008, 5372, 5356, 4936, 4220, 3316, 2392, 1556, 904, 456, 188, 60, 12, 1

Offset: 0

Geoffrey Critzer, Feb 19 2025

Let p = ({b_1},{b_2},...,{b_m}) be an ordered set partition of [n] into m blocks for some m, 1<=m<=n. A descent in p is an ordered pair (x,y) in [n]X[n] such that x is in b_i, y is in b_j, iy.
T(n,binomial(n,2)) = 1 (counts the ordered set partition ({n},{n-1},...,{2},{1})).
For n>=1, T(n,0) = 2^(n-1).
Sum_{k>=0} T(n,k)*2^k = A289545(n).
Sum_{k>=0} T(n,k)*3^k = A347841(n).
Sum_{k>=0} T(n,k)*4^k = A347842(n).
Sum_{k>=0} T(n,k)*5^k = A347843(n).
Sum_{k>=0} T(n,k)*6^k = A385408(n).
Sum_{k>=0} T(n,k)*7^k = A347844(n).
Sum_{k>=0} T(n,k)*8^k = A347845(n).
Sum_{k>=0} T(n,k)*9^k = A347846(n).
T(n,k) is the number of preferential arrangements of n labeled elements with exactly k inversions. For example, there 4 preferential rearrangements of length 3 with 1 inversion: 132, 213, 212, 131. - Kyle Celano, Aug 18 2025

			Triangle T(n,k) begins:
  1;
  1;
  2,  1;
  4,  4,  4,  1;
  8, 12, 18, 18,  12,  6,  1;
 16, 32, 60, 84, 100, 92, 76, 48, 24, 8, 1;
 ...

Alois P. Heinz, Rows n = 0..50, flattened
Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 2019-2020.
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025. See Theorem 1.

Columns k=0-2 give: A011782, A001787(n-1) for n>=1, 2*A268586.
Cf. A000670 (row sums), A008302 (the cases where each block has size 1).
Cf. A125810, A161680, A240796, A289545, A347841, A347842, A347843, A347844, A347845, A347846, A385408.

b:= proc(o, u, t) option remember; expand(`if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(x^(u+j-1)*b(o-j, u+j-1, 1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Feb 21 2025

nn = 7; B[n_] := FunctionExpand[QFactorial[n, u]]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[1/(1 -(e[z] - 1)), {z, 0, nn}], z]] // Grid

Sum_{k=0..binomial(n,2)} k * T(n,k) = A240796(n). - Alois P. Heinz, Feb 20 2025
T(n,k) is the coefficient of q^k in n!q times the coefficient of x^n in 1/(1- x - x^2/2!_q - x^3/3!_q - ...), where n!_q= 1*(1+q)*(1+q+q^2)*...*(1+q+...+q^(n-1)). - _Ira M. Gessel, Jun 24 2025
T(n,k) = Sum_{w} 2^(asc(w)), where w runs through the set of permutations with k inversions and asc(w) is the number of ascents of w. - Kyle Celano, Aug 18 2025

A381192 Irregular triangle read by rows. Properly color the vertices of a simple labeled graph on [n] using exactly n colors c_1=0, 0<=k<=binomial(n,2).

Original entry on oeis.org

1, 1, 3, 1, 21, 19, 7, 1, 315, 516, 419, 208, 65, 12, 1, 9765, 24186, 31445, 27488, 17538, 8420, 3050, 816, 153, 18, 1, 615195, 2080323, 3769767, 4754751, 4592847, 3555479, 2257723, 1188595, 519745, 187705, 55237, 12941, 2325, 301, 25, 1

Offset: 0

Geoffrey Critzer, Feb 16 2025

A descent in a labeled directed graph is an edge s->t such that s>t.
T(n,0) = A005329(n).
Sum_{k>=0} T(n,k)*k = A005329(n)*n(n-1)/8.

			     1;
     1;
     3,     1;
    21,    19,     7,     1;
   315,   516,   419,   208,    65,   12,   1;
  9765, 24186, 31445, 27488, 17538, 8420, 3050, 816, 153, 18, 1;
  ...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

CF. A005329, A381058, A011266 (row sums), A381102.

nn = 6; B[n_] :=FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1 + y)^Binomial[n, 2]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/(1 - z), {z, 0, nn}], z] /. y -> 1] // Grid

cp's OEIS Frontend

User: Geoffrey Critzer

A385355 Triangular array read by rows: T(n,k) is the number of n X n matrices A over GF(2) such that the dimension of the null space of A^n is equal to k, n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A384038 Number of 2n X 2n matrices M over GF(2) such that the column space of M is equal to the null space of M.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A383656 Triangular array read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k connected components, n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A383655 Triangle read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k isolated points, n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A382223 Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u=0, k>=0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A382363 Rectangular array read by antidiagonals, T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] such that for all u,v in [n], u->v implies u<=v and c(u)<=c(v), n>=0, k>=0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Views

Author

Keywords

Comments