A037306 -id:A037306 - cp's OEIS frontend

A183110 Period-length of the ultimate periodic behavior of the orbit of a list [1,1,1,...,1] of n 1's under the mapping defined in the comments.

1, 2, 1, 3, 3, 1, 4, 4, 4, 1, 5, 5, 5, 5, 1, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 8, 8, 8, 8, 8, 8, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1, 13, 13

Offset: 1

John W. Layman, Feb 01 2011

We use the list mapping introduced in A092964, whereby one removes the first term of the list, z(1), and adds 1 to each of the next z(1) terms (appending 1's if necessary) to get a new list.

This is also conjectured to be the length of the longest cycle of pebble-moves among the partitions of n (cf. A201144). - Andrew V. Sutherland

			Under the indicated mapping the list [1,1,1,1,1,1,1] of seven 1's results in the orbit [1,1,1,1,1,1,1], [2,1,1,1,1,1], [2,2,1,1,1], [3,2,1,1], [3,2,2], [3,3,1], [4,2,1], [3,2,1,1], ... which is clearly periodic with period-length 4, so a(7) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A037306, A092964, A178572, A178574.

f[p_] := Module[{pp, x, lpp, m, i}, pp = p; x = pp[[1]]; pp = Delete[pp,1]; lpp = Length[pp]; m = Min[x, lpp]; For[i = 1, i ≤ m, i++, pp[[i]]++]; For[i = 1, i ≤ x - lpp, i++, AppendTo[pp, 1]]; pp]; orb[p_] := Module[{s, v}, v = p; s = {v}; While[! MemberQ[s, v = f[v]], AppendTo[s, v]]; s]; attractor[p_] := Module[{orbp, pos, len, per}, orbp = orb[p]; pos = Flatten[Position[orbp, f[orbp[[-1]]]]][[1]] - 1; (*pos = steps to enter period*) len = Length[orbp] - pos; per = Take[orbp, -len]; Sort[per]]; a = {}; For[n = 1, n ≤ 80, n++, {rn = Table[1, {k, 1, n}]; orbn = orb[rn]; lenorb = Length[orbn]; lenattr = Length[attractor[rn]]; AppendTo[a, lenattr]}]; Print[a];

It appears, but has not yet been proved, that a(n)=1 if n=t(k) and a(n)=k if t(k-1) < n < t(k) where t(k) is the k-th triangular number t(k) = k*(k+1)/2.

A188160 For an unordered partition of n with k parts, remove 1 from each part and append the number k to get a new partition until a partition is repeated. a(n) gives the maximum steps to reach a period considering all unordered partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 12, 12, 12, 13, 18, 20, 20, 17, 18, 21, 28, 30, 30, 24, 24, 25, 32, 40, 42, 42, 35, 31, 32, 36, 45, 54, 56, 56, 48, 40, 40, 41, 50, 60, 70, 72, 72, 63, 54, 49, 50, 55, 66, 77, 88, 90, 90, 80, 70, 60, 60, 61

Offset: 1

Paul Weisenhorn, Mar 28 2011

nonn

Alternatively, if one iteratively removes the largest part z(1) and adds 1 to the next z(1) parts to get a new partition until a partition recurs, one gets the same maximum number of steps to reach a period.
The two shuffling operations are isomorphic for unordered partitions.
The two operations have the same length and number of periods for ordered and unordered partitions.
The steps count the operations including any pre-periodic part up to the end of first period, that is, the number of distinct partitions without including the first return.

			For k=6 and 0 <= j <= 1:
a(19)=21;  a(20)=28;  a(21)=30;  a(22)=30;  a(23)=24;  a(24)=24;  a(25)=25.
For n=4: (1+1+1+1)->(4)->(3+1)->(2+2)->(2+1+1)--> a(4)=4.
For n=5: (1+1+1+1+1)->(5)->(4+1)->(3+2)->(2+2+1)->(3+1+1)-->a(5)=5.

R. Baumann LOG IN, 4 (1987)
Halder, Heise Einführung in Kombinatorik, Hanser Verlag (1976) 75 ff.

Ethan Akin, Morton Davis, Bulgarian solitaire, Am. Math. Monthly 92 (4) (1985) 237-250
J. Brandt, Cycles of partitions, Proc. Am. Math. Soc. 85 (3) (1982) 483-486

Cf. A185700, A177922, A184996, A037306.

A188160 := proc(n)
        local k,j,T ;
        if n <= 2 then
                return n-1 ;
        end if;
        for k from 0 do
                T := k*(k+1) /2 ;
                if n = T and k >= 1 then
                        return k*(k-1) ;
                end if;
                if k>=4 then
                        j := T-1-n ;
                        if j>= 0 and j <= (k-4)/2 then
                                return k^2-k-2-(k+1)*j ;
                        end if;
                        j := n-T-1 ;
                        if j>= 0 and j <= (k-4)/2 then
                                return k^2-k-k*j ;
                        end if;
                end if;
                if k >= 2 then
                        j := n-(k^2+2*k-(k mod 2))/2 ;
                        if j>=0 and j <= 1 then
                                return (k^2+2*k-(k mod 2))/2+j
                        end if;
                end if;
        end do:
        return -1 ;
end proc: # R. J. Mathar, Apr 22 2011

a((k^2+k-2)/2-j) = k^2-k-2-(k+1)*j with 0<=j<=(k-4)/2  and 4<=k.
a((k^2+k+2)/2+j) = k^2-k-k*j with 0<=j<=(k-4)/2 and 4<=k,
a((k^2+2*k-(k mod 2))/2+j) = (k^2+2*k-(k mod 2))/2+j  with 0 <= j <= 1 and 2 <= k.
a(T(k)) = 2*T(k-1) = k^2-k with 1 <= k for the triangular numbers T(k)=A000217(k).

A178572 Numbers with ordered partitions that have periods of length 5.

Original entry on oeis.org

11, 47, 108, 194, 305, 441, 602, 788, 999, 1235, 1496, 1782, 2093, 2429, 2790, 3176, 3587, 4023, 4484, 4970, 5481, 6017, 6578, 7164, 7775, 8411, 9072, 9758, 10469, 11205, 11966, 12752, 13563, 14399, 15260, 16146, 17057, 17993, 18954, 19940, 20951

Offset: 1

Paul Weisenhorn, Dec 24 2010

From each ordered partition of the numbers (10+j) with 0

The a(n) sequence begins with 11 and each member has 1 period; the b(n) = A022282(n) sequence begins with 12 and each member has 2 periods; the c(n) = A022283(n) sequence begins with 13 and each member has 2 periods; the d(n) = n*(25*n + 3)/2 sequence begins with 14 and each member has 1 period of length 5.

			For n=11 the period is [(4,3,2,1,1), (4,3,2,2), (4,3,3,1), (4,4,2,1), (5,3,2,1)].
For n=47 the period is [(9,8,7,6,6,4,3,2,1,1), (9,8,7,7,5,4,3,2,2), (9,8,8,6,5,4,3,3,1), (9,9,7,6,5,4,4,2,1), (10,8,7,6,5,5,3,2,1)].
For n=12 the 2 periods are [(4,3,2,2,1), (4,3,3,2), (4,4,3,1), (5,4,2,1), (5,3,2,1,1)] and [(4,3,3,1,1), (4,4,2,2), (5,3,3,1), (4,4,2,1,1), (5,3,2,2)].
For n=49 the 2 periods are [(9,8,7,7,6,4,3,2,2,1), (9,8,8,7,5,4,3,3,2), (9,9,8,6,5,4,4,3,1), (10,9,7,6,5,5,4,2,1), (10,8,7,6,6,5,3,2,1,1)] and [(9,8,8,6,6,4,3,3,1,1), (9,9,7,7,5,4,4,2,2),(10,8,8,6,5,5,3,3,1), (9,9,7,6,6,4,4,2,1,1), (10,8,7,7,5,5,3,2,2)].

G. C. Greubel, Table of n, a(n) for n = 1..1000
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A092964, A037306.

List([1..50], n -> n*(25*n-3)/2); # G. C. Greubel, Jan 30 2019

[n*(25*n-3)/2: n in [1..50]]; // G. C. Greubel, Jan 30 2019

LinearRecurrence[{3,-3,1},{11,47,108},50] (* Harvey P. Dale, Jan 14 2019 *)
Table[n*(25*n-3)/2, {n,1,50}] (* G. C. Greubel, Jan 30 2019 *)

a(n)=n*(25*n-3)/2 \\ Charles R Greathouse IV, Jun 18 2017

[n*(25*n-3)/2 for n in (1..50)] # G. C. Greubel, Jan 30 2019

G.f. for a(n): (11 + 14*x)/(1-x)^3.
     for b(n): (12 + 13*x)/(1-x)^3.
     for c(n): (13 + 12*x)/(1-x)^3.
     for d(n): (14 + 11*x)/(1-x)^3.
All sequences have the same recurrence
  s(n+3) = 3*s(n+2) - 3*s(n+1) + s(n)
  with s(0)=0, s(1) = 10 + j, s(2) = 45 + 2*j and 0s(n) = n*(25*n - 5 + 2*j)/2  and 0The general formula for numbers with periods of length k:  a(k,j,n) = n*(k^2*n - k + 2*j)/2 with 0  For j=1 and j=(k-1) the numbers have 1 period.
  For 1A092964(k-4,j-1) periods.
  G.f.: (binomial(k,2)*(1+x) + j + (k-j)*x)/(1-x)^3.

A184996 For each ordered partition of n with k numbers, remove 1 from each part and add the number k to get a new partition, until a partition is repeated. Among all ordered partitions of n, a(n) gives the maximum number of steps needed to reach a period.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 11, 13, 15, 15, 16, 17, 22, 24, 24, 22, 23, 26, 33, 35, 35, 29, 30, 31, 38, 46, 48, 48, 41, 38, 39, 43, 52, 61, 63, 63, 55, 47, 48, 49, 58, 68, 78, 80, 80, 71, 62, 58, 59, 64, 75, 86, 97, 99, 99, 89, 79, 69, 70, 71, 82, 94, 106, 118, 120, 120, 109, 98, 87

Offset: 1

Paul Weisenhorn, Mar 28 2011

nonn

If one plays with p(n,n) unordered partitions, one gets the same number and length of periods.

If one removes the first part z(1) of each partition and adds 1 to the next z(1) parts to get a new partition, until a partition is repeated, one gets the same length and number of periods, playing with 2^(n-1) ordered or p(n,n) unordered partitions (A185700, A092964, A037306)

			For k=6: a(19)=26; a(20)=3; a(21)=35; a(22)=35; a(23)=29; a(24)=30; a(25)=31.
For n=4: (1+1+1+1)->(4)->(3+1)->(2+2)->(1+1+2)->(1+3)--> a(4)=5 steps.
For n=5: (1+1+1+1+1)->(5)->(4+1)->(3+2)->(2+1+2)->(1+1+3)->(2+3)->(1+2+2)--> a(5)=7 steps.

R. Baumann, Computer-Knobelei, LOGIN, 4 (1987), pages ?.
H. R. Halder and W. Heise, Einführung in Kombinatorik, Hanser Verlag, Munich, 1976, pp. 75ff.

Cf. A185700, A092964, A037306.

a((k^2+k-2)/2-j)=k^2-3-(k+1)*j with 0<=j<=(k-4) div 2 and 4<=k.
a((k^2+k+2)/2+j)=k^2-1-k*j with 0<=j<=(k-5) div 2 and 5<=k.
a((k^2+2*k-2+k mod 2)/2+j)=(k^2+4*k-2+k mod 2)/2+j with 0<=j<=2-k mod 2 and 4<=k.
a(T(k))=k^2-1 with 1<= k  for all triangular numbers T(k).

Partially edited by N. J. A. Sloane, Apr 08 2011

A211352 Refined triangle A211356: T(n,k) is the number of partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 4, 9, 3, 10, 1, 5, 3, 3, 1, 1, 3, 5, 15, 5, 30, 3, 15, 15, 10, 1, 15, 3, 5, 1, 1, 4, 7, 29, 10, 70, 7, 56, 54, 37, 4, 105, 21, 35, 1, 18, 29, 37, 4, 7, 7, 1

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,7,12,43... (A084423)
Row maxima: 1,1,1,2,3,10,30,105,420,1268,6300...
Distinct entries per row: 1,1,1,2,3,6,6,13,17,25,25...
Rightmost columns are those from the triangle of circular binomial coefficients A047996 without the second column (i.e.triangle A037306).

Tilman Piesk, Rows n=1..11 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211356, A000041, A084423, A047996, A037306.

A211353 Refined triangle A211357: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 4, 6, 3, 5, 1, 2, 1, 1, 1, 1, 3, 5, 10, 5, 15, 3, 5, 6, 3, 1, 3, 1, 1, 1, 1, 4, 7, 19, 10, 35, 7, 19, 21, 12, 4, 21, 7, 7, 1, 3, 4, 4, 1, 1, 1, 1, 1, 4, 10, 28, 14, 70, 14, 48, 56, 28, 10

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,6,10,28... (A054357)
Row maxima: 1,1,1,2,2,6,15,35,84,252,630,1542...
Distinct entries per row: 1,1,1,2,2,6,6,9,11,17,17,30...
Rightmost columns are those from the triangle of circular binomial coefficients A047996 without the second column (i.e.triangle A037306).

Tilman Piesk, Rows n=1..12 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211357, A000041, A054357, A047996, A037306.

A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 9, 7, 4, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 5, 12, 21, 26, 21, 12, 5, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 6, 19, 42, 66, 77, 66, 42, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Jan 22 2019

For n >= 1, T(n,k) is the number of nodes in n-dimensional space for Mysovskikh's cubature formula which is exact for any polynomial of degree k of n variables.

			Array begins:
  1, 1, 1,  1,  1,   1,   1,    1,    1,    1, ...
  1, 1, 2,  2,  3,   3,   4,    4,    5,    5, ...
  1, 1, 2,  4,  5,   7,  10,   12,   15,   19, ...
  1, 1, 3,  5,  9,  14,  21,   30,   42,   55, ...
  1, 1, 3,  7, 14,  26,  42,   66,   99,  143, ...
  1, 1, 4, 10, 21,  42,  77,  132,  215,  334, ...
  1, 1, 4, 12, 30,  66, 132,  246,  429,  715, ...
  1, 1, 5, 15, 42,  99, 215,  429,  805, 1430, ...
  1, 1, 5, 19, 55, 143, 334,  715, 1430, 2702, ...
  1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  5,  3,  1,  1;
  1, 4,  7,  9,  7,  4,  1,  1;
  1, 4, 10, 14, 14, 10,  4,  1, 1;
  1, 5, 12, 21, 26, 21, 12,  5, 1, 1;
  1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
  ...

Ronald Cools, Encyclopaedia of Cubature Formulas
Ivan P. Mysovskikh, On the construction of cubature formulae for very simple domains, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.

Main diagonal: A000108.

Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.

b(n, k) := (n + k)!/((n + 1)!*k!)$
T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
create_list(T(k, n - k), n, 0, 15, k, 0, n);

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A183110 Period-length of the ultimate periodic behavior of the orbit of a list [1,1,1,...,1] of n 1's under the mapping defined in the comments.

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A188160 For an unordered partition of n with k parts, remove 1 from each part and append the number k to get a new partition until a partition is repeated. a(n) gives the maximum steps to reach a period considering all unordered partitions of n.

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A178572 Numbers with ordered partitions that have periods of length 5.

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A184996 For each ordered partition of n with k numbers, remove 1 from each part and add the number k to get a new partition, until a partition is repeated. Among all ordered partitions of n, a(n) gives the maximum number of steps needed to reach a period.

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A211352 Refined triangle A211356: T(n,k) is the number of partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

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A211353 Refined triangle A211357: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

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A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

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Maxima