A293844 -id:A293844 - cp's OEIS frontend

A293845 Triangle read by rows: T(n,k) is the number of chains of length k in the partially ordered (by subspace inclusion) set of all subspaces of GF(2)^n, n>=0, 0<=k<=n.

1, 2, 1, 5, 7, 3, 16, 50, 56, 21, 67, 446, 1010, 945, 315, 374, 5395, 22692, 40455, 32550, 9765, 2825, 92881, 704601, 2167179, 3193155, 2255715, 615195, 29212, 2350136, 32061404, 162602418, 394534644, 496062000, 312519060, 78129765, 417199, 89342600, 2220570872, 18194735010, 68980503390, 138302085600, 151794972000

Offset: 0

Geoffrey Critzer, Oct 17 2017

			Triangle begins:
1;
2, 1;
5, 7, 3;
16, 50, 56, 21;
67, 446, 1010, 945, 315;
374, 5395, 22692, 40455, 32550, 9765;
...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

Cf. A289546, A293844 (row sums), A005329 (main diagonal), A006116 (column k = 0).

nn = 10; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,
  Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
     nn}] CoefficientList[Series[ eq[z]^2/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}]]]

T(n,k)/A005329(n) is the coefficient of y^k*x^n in eq(x)^2/(1 - y (eq(x) - 1)) where eq(x) is the q-exponential function.

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

cp's OEIS Frontend

A293845 Triangle read by rows: T(n,k) is the number of chains of length k in the partially ordered (by subspace inclusion) set of all subspaces of GF(2)^n, n>=0, 0<=k<=n.

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