A381426 -id:A381426 - cp's OEIS frontend

A289545 Number of flags in an n-dimensional vector space over GF(2).

1, 1, 4, 36, 696, 27808, 2257888, 369572160, 121459776768, 79991977040128, 105466641591287296, 278244130564826548224, 1468496684404408240109568, 15502543140842029367582248960, 327332729703063815298568073396224, 13823536566775628445052117519260598272

Offset: 0

Geoffrey Critzer, Jul 28 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..80
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A381299.

Column k=2 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(2^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..16);  # Alois P. Heinz, Feb 21 2025

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[ 1/(1 - (eq[z] - 1)) /. q -> 2, {z, 0, nn}], z]

a(n) = Sum A005329(n)/( A005329(n_1)*A005329(n_2)*...*A005329(n_k) ) where the sum is over all compositions of n = n_1 + n_2 + ... + n_k.
G.f. a(n)/A005329(n) is the coefficient of x^n in 1/(2 - eq(x)) where eq(x) is the 2-exponential function.
a(n) = Sum_{k=0..binomial(n,2)} 2^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

A381369 A(n,k) is the sum over all partitions of [n] of k^j for a partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 28, 52, 32, 1, 1, 2, 8, 47, 204, 203, 64, 1, 1, 2, 9, 72, 628, 2344, 877, 128, 1, 1, 2, 10, 103, 1552, 17327, 43160, 4140, 256, 1, 1, 2, 11, 140, 3276, 84416, 1022983, 1291952, 21147, 512

Offset: 0

Alois P. Heinz, Feb 21 2025

			Square array A(n,k) begins:
   1,   1,    1,     1,     1,      1,      1, ...
   1,   1,    1,     1,     1,      1,      1, ...
   2,   2,    2,     2,     2,      2,      2, ...
   4,   5,    6,     7,     8,      9,     10, ...
   8,  15,   28,    47,    72,    103,    140, ...
  16,  52,  204,   628,  1552,   3276,   6172, ...
  32, 203, 2344, 17327, 84416, 307867, 915848, ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened
Wikipedia, Inversion
Wikipedia, Partition of a set

Columns k=0-5 give: A011782, A000110, A125812, A125813, A125814, A125815.
Main diagonal gives A381373.
Cf. A125810, A381426.

b:= proc(o, u, t, k) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2, k), 0)+add(k^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1), k), j=`if`(t=0, 1, 1..o)))
    end:
A:= (n, k)-> b(n, 0$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Unprotect[Power]; 0^0 = 1; Protect[Power];
b[o_, u_, t_, k_] := b[o, u, t, k] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0, k], 0] + Sum[k^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1], k], {j, If[t == 0, {1}, Range[o]]}]];
A[n_, k_] := b[n, 0, 0, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

A(n,k) = Sum_{j>=0} k^j * A125810(n,j).

A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.

Original entry on oeis.org

1, 1, 5, 79, 3851, 567733, 251790113, 335313799327, 1340040415899803, 16067553466179577453, 577986341168068075687337, 62375143109859674070751394743, 20194282336027244435564571244298243, 19614041602745899032342581715038226919285

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 79 = 1 * 1 + 13 * 2 + 52 * 1, counting:
the unrefined chain 0 < (F_3)^3;
13 chains 0 < V < (F_3)^3, with dim(V) = 1; another
13 chains 0 < V < (F_3)^3, with dim(V) = 2; and
52 chains 0 < V_1 < V_2 < (F_3)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..64 (terms n=1..40 from Álvar Ibeas)

Cf. A289545, A347486, A036038, A381299.

Column k=3 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(3^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 21 2025

a(n) = Sum_{L partition of n} A347486(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(n) = Sum_{k=0..binomial(n,2)} 3^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

a(0)=1 prepended by Alois P. Heinz, Feb 21 2025

A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.

Original entry on oeis.org

1, 1, 6, 148, 14808, 5942608, 9546508128, 61355108818240, 1577381936031968640, 162213856617581098030336, 66726795842176170072717129216, 109792555585903911536355551233758208, 722612693482570097701467493432061846673408, 19023844570798442009810731239392846416136188284928

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 148 = 1 * 1 + 21 * 2 + 105 * 1, counting:
the unrefined chain 0 < (F_4)^3;
21 chains 0 < V < (F_4)^3, with dim(V) = 1; another
21 chains 0 < V < (F_4)^3, with dim(V) = 2; and
105 chains 0 < V_1 < V_2 < (F_4)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..58 (terms n = 1..40 from Álvar Ibeas)

Cf. A289545, A347487, A036038, A381299.

Column k=4 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(4^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 21 2025

a(n) = Sum_{L partition of n} A347487(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(n) = Sum_{k=0..binomial(n,2)} 4^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

a(0)=1 prepended by Alois P. Heinz, Feb 21 2025

A347843 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_5)^n.

Original entry on oeis.org

1, 1, 7, 249, 44643, 40065301, 179833594207, 4036127700341649, 452932494435315724443, 254139954749268142006053901, 712988623255130761190069046824407, 10001434425838325885839124865408303623049, 701474672607858244757589244286886103482442884243

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 249 = 1 * 1 + 31 * 2 + 186 * 1, counting:
the unrefined chain 0 < (F_5)^3;
31 chains 0 < V < (F_5)^3, with dim(V) = 1; another
31 chains 0 < V < (F_5)^3, with dim(V) = 2; and
186 chains 0 < V_1 < V_2 < (F_5)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..54 (terms n = 1..40 from Álvar Ibeas)

Cf. A289545, A347488, A036038, A381299.

Column k=5 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(5^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..16);  # Alois P. Heinz, Feb 21 2025

a(n) = Sum_{L partition of n} A347488(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(n) = Sum_{k=0..binomial(n,2)} 5^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

a(0)=1 prepended by Alois P. Heinz, Feb 21 2025

A347844 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_7)^n.

Original entry on oeis.org

1, 9, 571, 254451, 793949053, 17342194001493, 2651651342949844915, 2838097493373982791359403, 21263575996354049430406053548413, 1115175826296921591259255638404605916661, 409400558616252174531470888061523725583206190339

Offset: 1

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 571 = 1 * 1 + 57 * 2 + 456 * 1, counting:
the unrefined chain 0 < (F_7)^3;
57 chains 0 < V < (F_7)^3, with dim(V) = 1; another
57 chains 0 < V < (F_7)^3, with dim(V) = 2; and
456 chains 0 < V_1 < V_2 < (F_7)^3.

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

Cf. A289545, A347489, A036038.

Column k=7 of A381426.

a(n) = Sum_{L partition of n} A347489(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

A347845 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_8)^n.

Original entry on oeis.org

1, 1, 10, 804, 518376, 2674194448, 110368339035808, 36440751353074277952, 96254339565438079064819328, 2033964285682509941820879401890048, 343839935881726495233403720783311789640192, 465006794599984581603302662503095770372066384585728

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 804 = 1 * 1 + 73 * 2 + 657 * 1, counting:
the unrefined chain 0 < (F_8)^3;
73 chains 0 < V < (F_8)^3, with dim(V) = 1; another
73 chains 0 < V < (F_8)^3, with dim(V) = 2; and
657 chains 0 < V_1 < V_2 < (F_8)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..47 (terms n = 1..40 from Álvar Ibeas)

Cf. A289545, A347490, A036038.

Column k=8 of A381426.

a(n) = Sum_{L partition of n} A347490(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(0)=1 prepended by Alois P. Heinz, Jun 25 2025

A347846 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_9)^n.

Original entry on oeis.org

1, 1, 11, 1093, 979163, 7895396653, 572984959186643, 374244678702477629605, 2199939020346263706461674955, 116387990444553949414146511586296381, 55417662962428710787068813831544886356769891, 237482030708312867514661156730660313316831290472695733

Offset: 0

Álvar Ibeas, Sep 15 2021

nonn

			a(3) = 1093 = 1 * 1 + 91 * 2 + 910 * 1, counting:
the unrefined chain 0 < (F_9)^3;
91 chains 0 < V < (F_9)^3, with dim(V) = 1; another
91 chains 0 < V < (F_9)^3, with dim(V) = 2; and
910 chains 0 < V_1 < V_2 < (F_9)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..46 (terms n = 1..40 from Álvar Ibeas)

Cf. A289545, A347491, A036038, A381299.

Column k=9 of A381426.

a(n) = Sum_{L partition of n} A347491(n, L) * A036038(len(L), sig(L)), where sig(L) is the partition composed by the part multiplicities of L.

a(0)=1 prepended by Alois P. Heinz, Jun 25 2025

A381427 Sum over all ordered partitions of [n] of n^j for an ordered partition with j inversions.

Original entry on oeis.org

1, 1, 4, 79, 14808, 40065301, 2099255895008, 2651651342949844915, 96254339565438079064819328, 116387990444553949414146511586296381, 5327195120249449992420082364255283659438679552, 10333056290045508772052838892223597279253890797441054043823

Offset: 0

Alois P. Heinz, Feb 23 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..36
Wikipedia, Inversion
Wikipedia, Partition of a set

Main diagonal of A381426.

Cf. A062173, A135528, A381299, A381373.

b:= proc(o, u, t, k) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2, k), 0)+add(k^(u+j-1)*b(o-j, u+j-1, 1, k), j=1..o))
    end:
a:= n-> b(n, 0$2, n):
seq(a(n), n=0..11);

a(n) = Sum_{j=0..binomial(n,2)} n^j * A381299(n,j).
a(n) = A381426(n,n).
a(n) mod n = A062173(n) for n>=1.
a(n) mod 2 = A135528(n+1).

A385408 Sum over all ordered partitions of [n] of 6^j for an ordered partition with j inversions.

Original entry on oeis.org

1, 1, 8, 388, 113480, 199246816, 2099255895008, 132708276995157568, 50336523318422432038400, 114556539064849604787867141376, 1564256035642651626332994903500876288, 128158392280785912677966097933268099449960448, 62999559569114394473388668602373642996554916532377600

Offset: 0

Alois P. Heinz, Jun 27 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..51
Wikipedia, Inversion
Wikipedia, Partition of a set

Column k=6 of A381426.

Cf. A000670, A381299.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(6^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..14);

a(n) = Sum_{k=0..binomial(n,2)} A381299(n,k)*6^k.

cp's OEIS Frontend

A289545 Number of flags in an n-dimensional vector space over GF(2).

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A381369 A(n,k) is the sum over all partitions of [n] of k^j for a partition with j inversions; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A347841 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_3)^n.

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A347842 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_4)^n.

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A347843 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_5)^n.

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A347844 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_7)^n.

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A347845 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_8)^n.

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Extensions

A347846 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_9)^n.

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