A332716 -id:A332716 - cp's OEIS frontend

A332721 Number of compositions of n^2 with parts <= n, such that each element of [n] is used at least once.

1, 1, 3, 72, 5752, 1501620, 1171326960, 2571831080160, 15245263511750160, 236246829658682027760, 9325247205993698149853760, 917699267902161951609308035200, 221117091698491444413008381486903040, 128433050637127079872089064922773889126400

Offset: 0

Alois P. Heinz, Feb 21 2020

nonn

			a(2) = 3: 112, 121, 211.
a(3) = 72: 111123, 111132, 111213, 111231, 111312, 111321, 112113, 112131, 112311, 113112, 113121, 113211, 121113, 121131, 121311, 123111, 131112, 131121, 131211, 132111, 211113, 211131, 211311, 213111, 231111, 311112, 311121, 311211, 312111, 321111, 11223, 11232, 11322, 12123, 12132, 12213, 12231, 12312, 12321, 13122, 13212, 13221, 21123, 21132, 21213, 21231, 21312, 21321, 22113, 22131, 22311, 23112, 23121, 23211, 31122, 31212, 31221, 32112, 32121, 32211, 1233, 1323, 1332, 2133, 2313, 2331, 3123, 3132, 3213, 3231, 3312, 3321.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A000079, A000290, A103488, A332716, A332747, A332796, A373118.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (p+n)!
       /(n+1)!, add(b(n-i*j, i-1, p+j)/(j+1)!, j=0..n/i))
    end:
a:= n-> b(n*(n-1)/2, n$2):
seq(a(n), n=0..15);

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p + n)!/(n + 1)!, Sum[b[n - i*j, i - 1, p + j]/(j + 1)!, {j, 0, n/i}]];
a[n_] := b[n(n - 1)/2, n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

a(n) = A373118(n^2,n). - Alois P. Heinz, May 26 2024

A332796 Number of compositions of n^2 into parts >= n.

Original entry on oeis.org

1, 1, 2, 6, 26, 140, 882, 6349, 51284, 457704, 4459940, 47019819, 532485538, 6438774524, 82710138994, 1123798871990, 16090426592488, 241979954659728, 3811335657375786, 62712512310820402, 1075527196672980525, 19186234784992217621, 355349469934379290700

Offset: 0

Alois P. Heinz, Feb 24 2020

nonn

			a(0) = 1: (), the empty composition.
a(1) = 1: 1.
a(2) = 2: 22, 4.
a(3) = 6: 333, 36, 63, 45, 54, 9.
a(4) = 26: 4444, 556, 565, 655, 466, 646, 664, 457, 475, 547, 574, 745, 754, 448, 484, 844, 88, 79, 97, 6(10), (10)6, 5(11), (11)5, 4(12), (12)4, (16).

Alois P. Heinz, Table of n, a(n) for n = 0..170

Cf. A011782, A103488, A332716, A332721, A332747.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k), j=k..n))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k], {j, k, n}]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

A332747 Number of compositions of n^2, such that each element of [n] is used at least once as a part.

Original entry on oeis.org

1, 1, 3, 72, 6232, 1621620, 1241237520, 2675188471920, 15634073104902000, 239929277724680059440, 9411787539302194544158080, 922671287397731617736789070720, 221805878984619105095368813189002240, 128660270226206951104782827202740054476800

Offset: 0

Alois P. Heinz, Feb 21 2020

nonn

Some parts can be larger than n. Adding the condition that parts cannot be larger than n, we get A332721. Removing from A332721 the condition that each element of [n] has to be used, we get A332716.

			a(4) = 6232: all permutations of 4321111111, 432211111, 43222111, 4322221, 43321111, 4332211, 433321, 4432111, 443221, 543211, 64321.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A011782, A103488, A332716, A332721, A332796.

b:= proc(n, i, p, m) option remember; `if`(n=0, p!,
      `if`(i<1, 0, (t-> add(b(n-i*j, i-1, p+j, t)/(j+
      `if`(t=0, 1, 0))!, j=0..n/i))(`if`(i>m, m, 0))))
    end:
a:= n-> b(n*(n-1)/2$2, n$2):
seq(a(n), n=0..15);

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, p!,
     If[i < 1, 0, Function[t, [b[n - i*j, i - 1, p + j, t]/(j +
     If[t == 0, 1, 0])!, {j, 0, n/i}]][If[i > m, m, 0]]]];
a[n_] := b[n(n-1)/2, n(n-1)/2, n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 08 2021, after Alois P. Heinz *)

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A332721 Number of compositions of n^2 with parts <= n, such that each element of [n] is used at least once.

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A332796 Number of compositions of n^2 into parts >= n.

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A332747 Number of compositions of n^2, such that each element of [n] is used at least once as a part.

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