A289545 -id:A289545 - cp's OEIS frontend

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).

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

A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered 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, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 36, 75, 16, 1, 1, 6, 79, 696, 541, 32, 1, 1, 7, 148, 3851, 27808, 4683, 64, 1, 1, 8, 249, 14808, 567733, 2257888, 47293, 128, 1, 1, 9, 388, 44643, 5942608, 251790113, 369572160, 545835, 256, 1, 1, 10, 571, 113480, 40065301, 9546508128, 335313799327, 121459776768, 7087261, 512

Offset: 0

Alois P. Heinz, Feb 23 2025

			Square array A(n,k) begins:
   1,    1,       1,         1,          1,            1,             1, ...
   1,    1,       1,         1,          1,            1,             1, ...
   2,    3,       4,         5,          6,            7,             8, ...
   4,   13,      36,        79,        148,          249,           388, ...
   8,   75,     696,      3851,      14808,        44643,        113480, ...
  16,  541,   27808,    567733,    5942608,     40065301,     199246816, ...
  32, 4683, 2257888, 251790113, 9546508128, 179833594207, 2099255895008, ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
Wikipedia, Inversion
Wikipedia, Partition of a set

Columns k=0-9 give: A011782, A000670, A289545, A347841, A347842, A347843, A385408, A347844, A347845, A347846.
Main diagonal gives A381427.
Cf. A381299, A381369.

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, k)-> b(n, 0$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[o_, u_, t_, k_] := b[o, u, t, k] = If[u + o == 0, 1, If[t == 1, b[u + o, 0, 0, k], 0] + Sum[If[k == u + j - 1 == 0, 1, k^(u + j - 1)]*b[o - j, u + j - 1, 1, k], {j, 1, 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, Apr 19 2025, after Alois P. Heinz *)

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

A289546 Triangle read by rows. T(n,k) is the number of flags in an n dimensional vector space over GF(2) that have length exactly k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 14, 21, 0, 1, 65, 315, 315, 0, 1, 372, 4650, 13020, 9765, 0, 1, 2823, 87234, 527310, 1025325, 615195, 0, 1, 29210, 2291715, 27448764, 105413175, 156259530, 78129765, 0, 1, 417197, 88508205, 2043137265, 14019952275, 38897461575, 46487210175, 19923090075

Offset: 0

Geoffrey Critzer, Jul 28 2017

			Triangle begins:
  1;
  0, 1;
  0, 1,    3;
  0, 1,   14,    21;
  0, 1,   65,   315,    315;
  0, 1,  372,  4650,  13020,    9765;
  0, 1, 2823, 87234, 527310, 1025325, 615195;

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A005329 (main diagonal), A289545 (row sums).

nn = 8; eq[z_] := Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[Take[(Table[ FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,  nn}] CoefficientList[Series[ 1/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid

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

A293844 Number of chains in the partially ordered (by subspace inclusion) set of all subspaces of the vector space GF(2)^n.

Original entry on oeis.org

1, 3, 15, 143, 2783, 111231, 9031551, 1478288639, 485839107071, 319967908160511, 421866566365149183, 1112976522259306192895, 5873986737617632960438271, 62010172563368117470328995839, 1309330918812255261194272293584895, 55294146267102513780208470077042393087

Offset: 0

Geoffrey Critzer, Oct 17 2017

nonn

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

Row sums of A293845.

Cf. A005329, A289545.

nn = 16; 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 - (eq[z] - 1)) /. q -> 2, {z, 0, nn}], z]

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

cp's OEIS Frontend

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).

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A381426 A(n,k) is the sum over all ordered partitions of [n] of k^j for an ordered 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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A347846 a(n) is the number of (strict) chains of subspaces with ends 0 and (F_9)^n.

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