A373118
Number T(n,k) of compositions of n such that the set of parts is [k]; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 7, 0, 1, 11, 6, 0, 1, 20, 12, 0, 1, 32, 32, 0, 1, 54, 72, 0, 1, 87, 152, 24, 0, 1, 143, 311, 60, 0, 1, 231, 625, 180, 0, 1, 376, 1225, 450, 0, 1, 608, 2378, 1116, 0, 1, 986, 4566, 2544, 120, 0, 1, 1595, 8700, 5752, 360
Offset: 0
T(6,2) = 11: 1122, 1212, 1221, 2112, 2121, 2211, 11112, 11121, 11211, 12111, 21111.
T(7,3) = 12: 1123, 1132, 1213, 1231, 1312, 1321, 2113, 2131, 2311, 3112, 3121, 3211.
Triangle T(n,k) begins:
1;
0, 1;
0, 1;
0, 1, 2;
0, 1, 3;
0, 1, 7;
0, 1, 11, 6;
0, 1, 20, 12;
0, 1, 32, 32;
0, 1, 54, 72;
0, 1, 87, 152, 24;
0, 1, 143, 311, 60;
0, 1, 231, 625, 180;
0, 1, 376, 1225, 450;
0, 1, 608, 2378, 1116;
0, 1, 986, 4566, 2544, 120;
...
Cf.
A000124,
A000142,
A000217,
A000290,
A001710,
A003056,
A008289,
A332721,
A371417,
A373305,
A373306.
-
b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
`if`(i<1 or n b(n, k, 0):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..18);
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
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).
-
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 *)
A332716
Number of compositions of n^2 where each part is less than or equal to n.
Original entry on oeis.org
1, 1, 5, 149, 20569, 11749641, 26649774581, 236837126431501, 8237168505776637425, 1125036467745713090813969, 606147434557459526483161067501, 1293596348252277644272081532560154645, 10970544241076481629439275072320816659677161
Offset: 0
a(2) = 5: 22, 211, 121, 112, 1111.
-
b:= proc(n, k) option remember; `if`(n=0, 1,
add(b(n-d, k), d=1..min(n, k)))
end:
a:= n-> b(n^2, n):
seq(a(n), n=0..15);
-
b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - d, k], {d, 1, Min[n, k]}]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 31 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
a(4) = 6232: all permutations of 4321111111, 432211111, 43222111, 4322221, 43321111, 4332211, 433321, 4432111, 443221, 543211, 64321.
-
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 *)
A333048
Number of compositions of n^2 into powers of n.
Original entry on oeis.org
1, 1, 6, 20, 96, 572, 3971, 31201, 272334, 2605268, 27042522, 302171806, 3611295430, 45911641817, 618074912240, 8776287336812, 130994094465946, 2049114914257540, 33504826964461451, 571285301051283841, 10136481840545237652, 186803012671904648805
Offset: 0
a(0) = 1: the empty composition.
a(1) = 1: 1.
a(2) = 6: 1111, 112, 121, 211, 22, 4.
a(3) = 20: 111111111, 1111113, 1111131, 1111311, 1113111, 1131111, 1311111, 3111111, 11133, 11313, 11331, 13113, 13131, 13311, 31113, 31131, 31311, 33111, 333, 9.
-
a:= n-> `if`(n<2, 1, 1+add(binomial(n*(n-j)+j, j), j=0..n)):
seq(a(n), n=0..21);
Showing 1-5 of 5 results.
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