A381369 -id:A381369 - cp's OEIS frontend

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.

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).

A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

Original entry on oeis.org

1, 1, 2, 6, 28, 204, 2344, 43160, 1291952, 63647664, 5218320672, 719221578080, 168115994031040, 67159892835119296, 46166133463916209792, 54941957091151982047616, 113826217192695041078973184, 412563248965919999955196308224, 2627807814905396804499456018866688

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) is illustrated by:
a(2) = 1*(1) + 3*(1) + 1*(2) = 6;
a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;
a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.
Triangle A022166 begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 1; ...

Alois P. Heinz, Table of n, a(n) for n = 0..90

Cf. A022166, A125810, A125811, A125813, A125814, A125815.

Column k=2 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(2^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

a[0] = 1; a[n_] := a[n] = Sum[QBinomial[n-1, k, 2] a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

/* q-Binomial coefficients: */ {C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 2^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

Original entry on oeis.org

1, 1, 2, 7, 47, 628, 17327, 1022983, 132615812, 38522717107, 25526768401271, 39190247441314450, 141213238745969102393, 1207367655155905204747681, 24733467452839301566047854678, 1224709126636123500201799360630423, 147747406166666863538672620806542995763

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence: a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) is illustrated by:
  a(2) = 1*(1) + 4*(1) + 1*(2) = 7;
  a(3) = 1*(1) + 13*(1) + 13*(2) + 1*(7) = 47;
  a(4) = 1*(1) + 40*(1) + 130*(2) + 40*(7) + 1*(47) = 628.
Triangle A022167 begins:
  1;
  1, 1;
  1, 4, 1;
  1, 13, 13, 1;
  1, 40, 130, 40, 1;
  1, 121, 1210, 1210, 121, 1;
  1, 364, 11011, 33880, 11011, 364, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..72

Cf. A022167, A125810, A125811, A125812, A125814, A125815.

Column k=3 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(3^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[3^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

/* q-Binomial coefficients: */
C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 3^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A125815 q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5.

Original entry on oeis.org

1, 1, 2, 9, 103, 3276, 307867, 89520089, 83657942588, 258923776689771, 2717711483011792407, 98702105953049319472394, 12629828399521800714941435773, 5784963467206342855747483263957541, 9613516698678314330032600987632336641122, 58637855728567773833514895771659795097103477549

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence: a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 6*(1) + 1*(2) = 9;
a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;
a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.
Triangle A022169 begins:
  1;
  1,    1;
  1,    6,      1;
  1,   31,     31,       1;
  1,  156,    806,     156,      1;
  1,  781,  20306,   20306,    781,    1;
  1, 3906, 508431, 2558556, 508431, 3906, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A022169, A125810, A125811, A125812, A125813, A125814.

Column k=5 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(5^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[5^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

/* q-Binomial coefficients: */
{C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 5^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A125814 q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4.

Original entry on oeis.org

1, 1, 2, 8, 72, 1552, 84416, 12107584, 4726583424, 5150624868864, 16010990175691264, 144648776120641766400, 3857411545088966609514496, 307705704204270334224705015808, 74294186209325019487040708053442560, 54874536782175258883045772243829235417088

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

			The recurrence: a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) is illustrated by:
  a(2) = 1*(1) + 5*(1) + 1*(2) = 8;
  a(3) = 1*(1) + 21*(1) + 21*(2) + 1*(8) = 72;
  a(4) = 1*(1) + 85*(1) + 357*(2) + 85*(8) + 1*(72) = 1552.
Triangle A022168 begins:
  1;
  1,    1;
  1,    5,     1;
  1,   21,    21,      1;
  1,   85,   357,     85,     1;
  1,  341,  5797,   5797,   341,    1;
  1, 1365, 93093, 376805, 93093, 1365, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..65

Cf. A022168, A125810, A125811, A125812, A125813, A125815.

Column k=4 of A381369.

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(4^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[4^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

/* q-Binomial coefficients: */
{C_q(n,k)=if(n

a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 4^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.

Original entry on oeis.org

1, 1, 2, 7, 72, 3276, 915848, 2011878835, 42723411900032, 10608257527069388539, 35808039364308986083608352, 1828963737334508176477805993389490, 1618534282345584818909121118371843799592960, 28472613161534902071627567919297331348486838233018341

Offset: 0

Alois P. Heinz, Feb 21 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..41
Wikipedia, Inversion
Wikipedia, Partition of a set

Main diagonal of A381369.

Cf. A062173, A120325, A125810, A381427.

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-> b(n, 0$2, n):
seq(a(n), n=0..15);

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_] := b[n, 0, 0, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

a(n) = Sum_{j>=0} n^j * A125810(n,j).
a(n) = A381369(n,n).
a(n) mod n = A062173(n) for n>=1.
a(n) mod 2 = A120325(n+1) for n>=1.

cp's OEIS Frontend

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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A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

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A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

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A125815 q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5.

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A125814 q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4.

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A381373 Sum over all partitions of [n] of n^j for a partition with j inversions.

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