A289545 -id:A289545 - cp's OEIS frontend

A381102 Irregular triangle read by rows. For each j, 1<=j<=n properly color the vertices of a labeled graph on [n] using each of the first j colors in the color set {c1=0, 0<=k<=binomial(n,2).

1, 1, 4, 1, 36, 27, 9, 1, 696, 983, 731, 330, 93, 15, 1, 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 1, 2257888, 6803655, 11412586, 13504721, 12316799, 9026017, 5427090, 2700863, 1112555, 376459, 103002, 22203, 3619, 417, 30, 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) = A289545(n).

			     1;
     1;
     4,     1;
    36,    27,     9,     1;
   696,   983,   731,   330,    93,    15,    1;
 27808, 60615, 72662, 59113, 35197, 15731, 5269, 1287, 216, 22, 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. A334282(row sums), A289545, A381058, A381192.

nn = 5; 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 - (e[z] - 1)), {z, 0, nn}], z] /. y -> 1] // Grid

A333797 Total number of saturated chains in the lattices L_n(2) of subspaces (ordered by inclusion) of the vector space GF(2)^n.

Original entry on oeis.org

1, 3, 14, 114, 1777, 55461, 3496868, 444131448, 113253936439, 57872769803787, 59203843739029706, 121190268142727296926, 496274148044956457612893, 4064981546636275903297015089, 66596592678542112197488335080432, 2182170552297789390998576752287351492

Offset: 0

Geoffrey Critzer, Apr 05 2020

nonn

These are the chains counted in A293844 that are saturated. A chain C in poset P is saturated if there is no z in P - C such that x < z < y for some x,y in C and such that C union {z} is a chain.

Cf. A289545, A293844.

nn = 15; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];
Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[eq[z]^2/(1 - z) /. q -> 2, {z, 0, nn}], z]

a(n)/A005329(n) is the coefficient of x^n in eq(x)^2/(1 - x) where eq(x) is the q-exponential function.

a(n) ~ A299998 * 2^(n*(n+1)/2). - Vaclav Kotesovec, Apr 07 2020

A347847 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_11)^n.

Original entry on oeis.org

1, 13, 1863, 2939835, 51032042341, 9744480919457833, 20467619608367348894343, 472899301427912499856338638715, 120188439043897767650882899915804254541, 336007833514175643082885509533867247163277824513

Offset: 1

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 1863 = 1 * 1 + 133 * 2 + 1596 * 1, counting:
the unrefined chain 0 < (F_11)^3;
133 chains 0 < V < (F_11)^3, with dim(V) = 1; another
133 chains 0 < V < (F_11)^3, with dim(V) = 2; and
1596 chains 0 < V_1 < V_2 < (F_11)^3.

Álvar Ibeas, Table of n, a(n) for n = 1..40

Cf. A289545, A347492, A036038.

a(n) = Sum_{L partition of n} A347492(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

A348116 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_2)^n, counted up to coordinate permutation.

Original entry on oeis.org

1, 3, 13, 82, 747, 10248, 217703, 7530572, 447825441

Offset: 1

Álvar Ibeas, Oct 01 2021

Cf. A348113, A036038, A289545.

a(n) = Sum_{L partition of n} A348113(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

cp's OEIS Frontend

A381102 Irregular triangle read by rows. For each j, 1<=j<=n properly color the vertices of a labeled graph on [n] using each of the first j colors in the color set {c1=0, 0<=k<=binomial(n,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A333797 Total number of saturated chains in the lattices L_n(2) of subspaces (ordered by inclusion) of the vector space GF(2)^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A347847 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_11)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A348116 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_2)^n, counted up to coordinate permutation.

Views

Author

Keywords

Crossrefs

Formula