A120698 -id:A120698 - cp's OEIS frontend

A120695 Set partitions reversed interpreted as factorial base numbers.

0, 1, 3, 5, 9, 15, 11, 17, 23, 33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119, 153, 273, 177, 297, 417, 159, 279, 399, 183, 303, 423, 207, 327, 447, 567, 155, 275, 395, 179, 299, 419, 203, 323, 443, 563, 161, 281, 401, 185, 305, 425, 209, 329, 449

Offset: 0

Franklin T. Adams-Watters, Jun 28 2006

			The third set partition is {1,2}, 21 in base factorial is 5, so a(3) = 5.
Triangle begins:
   0;
   1;
   3,  5;
   9, 15, 11, 17, 23;
  33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119;

Alois P. Heinz, Rows n = 0..9, flattened

Cf. A000110 (row lengths), A120698, A120696 (sorted), A120697, A071155.

Column k=0 gives A007489.

b:= proc(n, m, t) option remember; `if`(n=0, [0], [seq(map(
       x-> x+j*t!, b(n-1, max(m, j), t+1))[], j=1..m+1)])
    end:
T:= n-> b(n, 0, 1)[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Apr 04 2016

A120696 Numbers whose factorial representation contains no zeros and each digit is at most one larger than the largest following digit.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 23, 33, 35, 39, 41, 47, 57, 59, 63, 65, 71, 83, 87, 89, 95, 119, 153, 155, 159, 161, 167, 177, 179, 183, 185, 191, 203, 207, 209, 215, 239, 273, 275, 279, 281, 287, 297, 299, 303, 305, 311, 323, 327, 329, 335, 359, 395, 399, 401, 407

Offset: 0

Franklin T. Adams-Watters, Jun 28 2006

0 is included by considering it to have the empty string as its factorial base representation.

Alois P. Heinz, Rows n = 0..9, flattened

Cf. A000110 (row lengths), A120698, A120695, A120697, A071156.

A085693 Let f(0) = 1, f(1) = 12, f(2) = 223, f(3) = 3334, = f(4) = 44445, f(5) = 555556, etc. Sequence gives concatenation of 0, f(0), f(f(0)), f(f(f(0))), ...

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 5, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3

Offset: 0

Philippe Deléham, Aug 14 2003; revised Sep 18 2005

			0;
1;
1, 2;
1, 2, 2, 2, 3;
1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4; ...
For large m, the m-th block is given by A085694.

Cf. A085694, A000110, A048993 (the number of occurrences of k in n-th block).

Row maxima of A120698.

Edited by Charles R Greathouse IV, Apr 30 2010

A376937 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct rhyme scheme from that of any other subsequence of the sequence thus far.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 3, 4, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 6, 6, 4, 5, 6, 4, 5, 6, 7, 5, 5, 5, 3, 4, 5, 6, 6, 6, 6, 4, 5, 6, 7, 8, 6, 6, 7, 5, 6, 7, 8, 7, 5, 4, 5, 6, 6, 5, 5, 6, 5, 4, 5, 5, 6, 5, 5, 5, 5, 6, 6, 5, 6, 7, 8, 6, 6, 7, 8, 5, 6, 7, 8, 9, 9

Offset: 1

Neal Gersh Tolunsky, Oct 11 2024

nonn

In other words, a(n) is the length of the shortest subsequence ending at a(n-1) which has a unique rhyme scheme among all rhyme schemes of subsequences of the sequence thus far. Alternatively, this is (the length of the longest subsequence ending at a(n-1) whose rhyme scheme has occurred before as that of another subsequence) plus 1.

The rhyme scheme of a subsequence is found by assigning 1 to the first unique value, 2 to the second unique value, 3 to the third, and so on.

			a(6)=4 because the length-4 subsequence a(2..5) = 1,2,2,3 has the shortest unique rhyme scheme (1,2,2,3 or A,B,B,C,) which does not occur elsewhere as the rhyme scheme of any other subsequence in the sequence thus far. No shorter subsequence ending in a(5) with a unique rhyme scheme exists in the sequence thus far. For example, a(6) cannot be 3 because the length-3 subsequence a(3..5) = 2,2,3 has the same rhyme scheme (1,1,2 or A,A,B) as that of the subsequence a(1..3) = 1,1,2.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Neal Gersh Tolunsky, Ordinal Transform of 500000 terms

Cf. A377079, A375207, A120698 (rhyme schemes), A000110.

from itertools import count, islice
def rs(t): # rhyme scheme of t
    s, k, m  = [], 1, dict()
    for e in t:
        if e not in m: m[e] = k; k += 1
        s.append(m[e])
    return tuple(s)
def agen(): # generator of terms
    a, R, maxL = [1], set(), 0  # maxL = max length of rhyme schemes stored
    for n in count(1):
        yield a[-1]
        for k in range(1, n+1):
            if k > maxL:  # must increase length of rhyme schemes stored
                maxL += 1
                R.update(rs(a[i:i+maxL]) for i in range(n-maxL))
            if rs(a[-k:]) not in R:
                break
        an = k
        R.update(rs(a[-i:]) for i in range(1, maxL+1))
        a.append(an)
print(list(islice(agen(), 90))) # Michael S. Branicky, Oct 13 2024

A120697 Set partitions interpreted as base m+1 numbers, where m is the size of the set.

Original entry on oeis.org

0, 1, 4, 5, 21, 22, 25, 26, 27, 156, 157, 161, 162, 163, 181, 182, 183, 186, 187, 188, 191, 192, 193, 194, 781, 782, 786, 787, 788, 806, 807, 808, 811, 812, 813, 816, 817, 818, 819, 906, 907, 908, 911, 912, 913, 916, 917, 918, 919, 931, 932, 933, 936, 937, 938

Offset: 0

Franklin T. Adams-Watters, Jun 28 2006

			Set partition number 3 is {1,2}, 12 in base 3 is 5, so a(3) = 5.

Cf. A000110 (row lengths), A120698, A120695.

A120699 Lengths of set partitions.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Franklin T. Adams-Watters, Jun 28 2006

nonn

Each n occurs Bell(n) times.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000110, A120698.

Flatten[Array[PadRight[{}, BellB[#], #]&, 7, 0]] (* Paolo Xausa, Feb 13 2024 *)

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Feb 18 2024

A193023 Triangle read by rows: the n-th row has length A000110(n) and contains all set partitions of an n-set in canonical order.

Original entry on oeis.org

1, 11, 12, 111, 112, 121, 122, 123, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1234, 11111, 11112, 11121, 11122, 11123, 11211, 11212, 11213, 11221, 11222, 11223, 11231, 11232, 11233, 11234, 12111, 12112, 12113, 12121

Offset: 1

N. J. A. Sloane, Jul 14 2011

The set partition of [1,2,3,4] given by 13/2/4 would be encoded as 1213: simply record which part i is in, for i=1..n.
To get row n, read row n-1 from left to right. If row n-1 contains a word abc...d, in which the maximal number is m, then in row n we place the words abc...d1, abc...d2, abc...d3, ..., abc...d(m+1).
This provides a canonical ordering for partitions of a labeled set.

			Triangle begins:
  1;
  11,12;
  111,112,121,122,123;
  1111,1112,1121,1122,1123,1211,1212,1213,1221,1222,1223,1231,1232,1233,1234;
  11111,11112,11121,11122,11123,...

Alois P. Heinz, Rows n = 1..8, flattened
R. Kaye, A Gray code for set partitions, Info. Proc. Letts., 5 (1976), 171-173.

This is different from A071159.

Cf. A000110, A120698.

b:= proc(n) option remember;
      `if`(n=1, [[1]], map(x-> seq([x[], i], i=1..max(x[])+1), b(n-1)))
    end:
T:= n-> map(x-> parse(cat(x[])), b(n))[]:
seq(T(n), n=1..5);  # Alois P. Heinz, Sep 30 2011

b[n_] := b[n] = If[n == 1, {{1}}, Table[Append[#, i], {i, 1, Max[#]+1}]& /@ b[n-1] // Flatten[#, 1]&];
T[n_] := FromDigits /@ b[n];
Array[T, 8] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

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A120695 Set partitions reversed interpreted as factorial base numbers.

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A120696 Numbers whose factorial representation contains no zeros and each digit is at most one larger than the largest following digit.

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A085693 Let f(0) = 1, f(1) = 12, f(2) = 223, f(3) = 3334, = f(4) = 44445, f(5) = 555556, etc. Sequence gives concatenation of 0, f(0), f(f(0)), f(f(f(0))), ...

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A376937 a(1)=1; thereafter a(n) is the smallest k for which the subsequence a(n-k..n-1) has a distinct rhyme scheme from that of any other subsequence of the sequence thus far.

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Python

A120697 Set partitions interpreted as base m+1 numbers, where m is the size of the set.

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A120699 Lengths of set partitions.

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Formula

A193023 Triangle read by rows: the n-th row has length A000110(n) and contains all set partitions of an n-set in canonical order.

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