cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A227955 Triangle read by rows, T(n, k) = prime(1)^p(k,1)*...*prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 8, 210, 60, 36, 24, 16, 2310, 420, 180, 120, 72, 48, 32, 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64, 510510, 60060, 13860, 6300, 9240, 2520, 1800, 1080, 1680, 720, 432, 480, 288, 192, 128, 9699690, 1021020, 180180, 69300, 44100
Offset: 0

Views

Author

Peter Luschny, Aug 01 2013

Keywords

Comments

The sequence can be seen as an encoding of Young's lattice (see the links).
The ordering of Young's lattice is such that for two Young diagrams s, t, we have s <= t if and only if the Young diagram of s fits entirely inside the Young diagram of t (when the two diagrams are arranged so their lower-left corners coincide.) This order translates to our encoding as the divisibility relation. The number corresponding to s divides the number corresponding to t if and only if s <= t.
The partition corresponding to a number can be recovered as the exponents of the primes in the prime factorization of the number.

Examples

			For instance the partitions of 4 are ordered [1,1,1,1], [2,1,1,0], [2,2,0,0], [3,1,0,0], [4,0,0,0]. Consider the partition P = (3,2,1,1) written as a Young diagram (in French notation):
    [ ]
    [ ]
    [ ][ ]
    [ ][ ][ ]
Next replace the boxes at the bottom line by the sequence of primes and write the number of boxes in the same column as exponents; then multiply. 2^4*3^2*5^1 = 720. 720 will appear in line 7 of the triangle (because P is a partition of 7) at position 10 (because the sequence of exponents [4, 2, 1] is the 10th partition in the order of partitions which we assume).
[0]     1,
[1]     2,
[2]     6,    4,
[3]    30,   12,    8,
[4]   210,   60,   36,  24,  16,
[5]  2310,  420,  180, 120,  72,  48,  32,
[6] 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64.

Links

Crossrefs

Reversed rows: A036035, row sums: A074140.

Programs

  • Maple
    with(combinat):
A227955_row := proc(n) local e, w, p;
p := [seq(ithprime(i), i=1..n)];
w := e -> mul(p[i]^e[nops(e)-i+1], i=1..nops(e));
seq(w(e), e = partition(n)) end:
seq(print(A227955_row(i)), i=0..8);
  • Sage
    def A227955_row(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        L.append(mul(P[i]^p[i] for i in range(len(p))))
    return L[::-1]
for n in (0..8): A227955_row(n)

A238954 Maximal size of an antichain in graded colexicographic order of exponents.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 7, 10, 1, 2, 3, 4, 4, 6, 7, 8, 10, 14, 20, 1, 2, 3, 4, 4, 6, 7, 8, 8, 11, 13, 15, 18, 25, 35, 1, 2, 3, 4, 5, 4, 6, 8, 9, 10, 8, 12, 14, 16, 19, 16, 22, 26, 30, 36, 50, 70, 1, 2, 3, 4, 5, 4, 6, 8, 9, 9, 11, 12, 8, 12, 15, 17, 19, 22, 16, 23, 26, 30, 35, 31, 41, 48, 56, 66, 91, 126
Offset: 0

Views

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 3, 4, 6;
  1, 2, 3, 4, 5, 7, 10;
  1, 2, 3, 4, 4, 6,  7, 8, 10, 14, 20;
  ...

Links

Crossrefs

Cf. A096825 in graded colexicographic order.
Cf. A036035, A238967.

Programs

  • PARI
    \\ here b(n) is A096825.
b(n)={my(h=bigomega(n)\2); sumdiv(n, d, bigomega(d)==h)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Apr 25 2020

Formula

T(n,k) = A096825(A036035(n,k)).

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 25 2020

A238956 Degree of divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 5, 5, 5, 2, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 2, 3, 4, 4, 4, 5, 5, 6, 5, 6, 7, 6, 7, 7, 7, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 5, 6, 6, 7, 8, 6, 7, 8, 7, 8, 8, 8, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 9, 7, 8, 9, 8, 9, 9, 9
Offset: 0

Views

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

Examples

			Triangle T(n,k) begins:
  0;
  1;
  2, 2;
  2, 3, 3;
  2, 3, 4, 4, 4;
  2, 3, 4, 4, 5, 5, 5;
  2, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6;
  ...

Links

Crossrefs

Cf. A238949 in graded colexicographic order.
Cf. A036035, A238969.

Programs

  • PARI
    C(sig)={sum(i=1, #sig, if(sig[i]>1, 2, 1))}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020

Formula

T(n,k) = A238949(A036035(n,k)).

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 01 2020

A238961 The size (the number of arcs) in the transitive closure of divisor lattice in graded colexicographic order.

Original entry on oeis.org

0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 84, 115, 156, 189, 238, 288, 438, 665, 28, 70, 108, 130, 165, 240, 268, 324, 365, 492, 594, 746, 900, 1362, 2059, 36, 92, 147, 186, 200, 224, 342, 410, 495, 552, 519, 750, 836, 1008, 1215, 1135, 1524, 1836, 2302, 2772, 4182, 6305
Offset: 0

Views

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

Examples

			Triangle T(n,k) begins:
   0;
   1;
   3,  5;
   6, 12, 19;
  10, 22, 27, 42,  65;
  15, 35, 48, 74,  90, 138, 211;
  21, 51, 75, 84, 115, 156, 189, 238, 288, 438, 665;
  ...

Links

Crossrefs

Cf. A238952 in graded colexicographic order.
Cf. A036035, A238974.

Programs

  • PARI
    \\ here b(n) is A238952.
b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Apr 25 2020

Formula

T(n,k) = A238952(A036035(n,k)).

Extensions

Offset changed and terms a(50) and beyond from Andrew Howroyd, Apr 25 2020

A238962 Number of perfect partitions in graded colexicographic order.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683, 64, 256, 544, 768, 976, 1888, 2316, 3172, 3408, 5740, 7880, 10404, 14300, 25988, 47293, 128, 576, 1376, 2208, 2568, 2496, 5536, 7968
Offset: 0

Views

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

Examples

			Triangle T(n,k) begins:
   1;
   1;
   2,   3;
   4,   8,  13;
   8,  20,  26,  44,  75;
  16,  48,  76, 132, 176, 308, 541;
  32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683;
  ...

Links

Crossrefs

Row sums are A035341.
Cf. A002033 in graded colexicographic order.
Cf. A036035, A074206, A238975.

Programs

  • PARI
    \\ here b(n) is A074206.
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
b(n)={if(!n, 0, my(sig=factor(n)[,2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k))))}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Aug 30 2020

Formula

T(n,k) = A074206(A036035(n,k)). - Andrew Howroyd, Apr 25 2020

Extensions

Offset changed and terms a(44) and beyond from Andrew Howroyd, Apr 25 2020

A339677 Partition array: T(n, k) is the number of aperiodic necklaces (Lyndon words) on a multiset of colored beads (of size n) whose color multiplicities form the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 1, 3, 6, 0, 1, 2, 4, 6, 12, 24, 0, 1, 2, 3, 5, 10, 14, 20, 30, 60, 120, 0, 1, 3, 5, 6, 15, 20, 30, 30, 60, 90, 120, 180, 360, 720, 0, 1, 3, 7, 8, 7, 21, 35, 51, 70, 42, 105, 140, 210, 312, 210, 420, 630, 840, 1260, 2520, 5040, 0, 1, 4, 9, 14, 8, 28, 56, 70, 84, 140
Offset: 1

Views

Author

Álvar Ibeas, Dec 12 2020

Keywords

Comments

As in A212359, A072605, and A261600, for each partition, the base set of beads is fixed.
Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order. We do accordingly for A036038 and A212359.

Examples

			Array begins:
  k:  1 2 3 4 5  6  7  8  9 10  11  12  13  14  15
      --------------------------------------------
n=1:  1
n=2:  0 1
n=3:  0 1 2
n=4:  0 1 1 3 6
n=5:  0 1 2 4 6 12 24
n=6:  0 1 2 3 5 10 14 20 30 60 120
n=7:  0 1 3 5 6 15 20 30 30 60  90 120 180 360 720
Consider partition L = (4, 2). There are 3 = A212359(6, L) necklaces on the bead set {a^4, b^2}: (aaaabb), (aaabab), and (aabaab). The latter has a period smaller than its size (3 < 6), whereas the other two are aperiodic. Hence, T(6, L) = 2.
T(n, (1,...,1)) = A212359(n, (1,...,1)) = (n-1)!, counting necklaces with n beads, each in a different color.

Links

Crossrefs

Cf. A036035, A036038, A072605, A185974, A212359, A261600 (row sums), A298941, A318808.

Programs

  • PARI
    C(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, moebius(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
Row(n)=[C(Vec(p)) | p<-partitions(n)]
for(n=1, 7, print(Row(n))) \\ Andrew Howroyd, Dec 14 2020

Formula

Let L be a partition of n and d be the gcd of its parts. Then,
T(n, L) = n^(-1) * Sum_{v|d} mu(v) * A036038(n/v, L/v), where L/v is the partition obtained from L after dividing each part by v.
T(n, L) = Sum_{v|d} mu(v) * A212359(n/v, L/v).
T(n, L) = n^(-1) * A036038(n, L) - Sum_{1
T(n,k) = A298941(A036035(n,k)) = A318808(A185974(n,k)). - Andrew Howroyd, Dec 14 2020
Previous Showing 31-36 of 36 results.

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