Rui Duarte - cp's OEIS frontend

A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

1, 4, 2, 24, 15, 9, 200, 136, 100, 64, 2160, 1535, 1215, 945, 625, 28812, 21036, 17286, 14406, 11526, 7776, 458752, 341103, 286671, 247296, 211456, 172081, 117649, 8503056, 6405904, 5464712, 4811528, 4251528, 3691528, 3038344, 2097152, 180000000, 136953279, 118078911, 105372819, 94921875, 85078125, 74627181, 61921089, 43046721

Offset: 1

Rui Duarte, Jan 22 2018

T(n,k) is the number of pairs (f,i) such that f is a parking function and f(i) = k.

			Triangle begins:
====================================================================
n\k|       1       2       3       4       5       6       7       8
---|----------------------------------------------------------------
1  |       1
2  |       4       2
3  |      24      15       9
4  |     200     136     100      64
5  |    2160    1535    1215     945     625
6  |   28812   21036   17286   14406   11526    7776
7  |  458752  341103  286671  247296  211456  172081  117649
8  | 8503056 6405904 5464712 4811528 4251528 3691528 3038344 2097152
  ...

Cf. A000169, A000272, A089946, A298592, A298594, A298597.

Table[n Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

T(n,k) = n*Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
T(n,k) = n*A298592(n,k).
T(n,k) = n*Sum_{j=k..n} A298594(n,j).
T(n,k) = Sum_{j=k..n} A298597(n,j).
Sum_{k=1..n} T(n,k) = n*A000272(n+1).
T(n+1,1) = A089946(n), T(n,n) = A000169(n). - Andrey Zabolotskiy, Feb 21 2018

A298594 Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 16, 9, 9, 16, 125, 64, 54, 64, 125, 1296, 625, 480, 480, 625, 1296, 16807, 7776, 5625, 5120, 5625, 7776, 16807, 262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144, 4782969, 2097152, 1411788, 1161216, 1093750, 1161216, 1411788, 2097152, 4782969

Offset: 1

Rui Duarte, Jan 22 2018

			Triangle begins:
       1;
       1,      1;
       3,      2,     3;
      16,      9,     9,    16;
     125,     64,    54,    64,   125;
    1296,    625,   480,   480,   625,  1296;
   16807,   7776,  5625,  5120,  5625,  7776,  16807;
  262144, 117649, 81648, 70000, 70000, 81648, 117649, 262144;
  ...

Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 6.

Cf. A000272, A298592, A298593, A298597.

Table[Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

T(n,k) = binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).
T(n,k) = A298592(n,k) - A298592(n,k+1).
T(n,k) = (A298593(n,k) - A298593(n,k+1))/n.
T(n,k) = A298597(n,k)/n.
T(n,1) = A000272(n+2).
T(n,n) = A000272(n+2).
T(n,k) = T(n,n-k).

A298597 Number T(n,k) of times the value k appears on the parking functions of length n and such that if we replace that value k with k+1 we don't get a parking function.

Original entry on oeis.org

1, 2, 2, 9, 6, 9, 64, 36, 36, 64, 625, 320, 270, 320, 625, 7776, 3750, 2880, 2880, 3750, 7776, 117649, 54432, 39375, 35840, 39375, 54432, 117649, 2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152, 43046721, 18874368, 12706092, 10450944, 9843750, 10450944, 12706092, 18874368, 43046721

Offset: 1

Rui Duarte, Jan 22 2018

			Triangle begins:
        1;
        2,      2;
        9,      6,      9;
       64,     36,     36,     64;
      625,    320,    270,    320,    625;
     7776,   3750,   2880,   2880,   3750,   7776;
   117649,  54432,  39375,  35840,  39375,  54432, 117649;
  2097152, 941192, 653184, 560000, 560000, 653184, 941192, 2097152;
  ...

Cf. A000169, A000272, A298592, A298593, A298594.

Table[n Binomial[n - 1, k - 1] k^(k - 2)*(n + 1 - k)^(n - 1 - k), {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

T(n,k) = n*binomial(n-1, k-1)*k^(k-2)*(n+1-k)^(n-1-k).
T(n,k) = n*A298594(n,k).
T(n.k) = A298593(n,k)-A298593(n,k+1).
T(n,k) = n*(A298592(n,k)-A298592(n,k+1)).
T(n,1) = n*A000272(n+2).
T(n,n) = n*A000272(n+2).
T(n,1) = A000169(n).
T(n,n) = A000169(n).
T(n,k) = T(n,n-k).

A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.

Original entry on oeis.org

1, 2, 1, 8, 5, 3, 50, 34, 25, 16, 432, 307, 243, 189, 125, 4802, 3506, 2881, 2401, 1921, 1296, 65536, 48729, 40953, 35328, 30208, 24583, 16807, 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144, 20000000, 15217031, 13119879, 11708091, 10546875, 9453125, 8291909, 6880121, 4782969

Offset: 1

Rui Duarte, Jan 22 2018

			Triangle begins:
        1;
        2,      1;
        8,      5,      3;
       50,     34,     25,     16;
      432,    307,    243,    189,    125;
     4802,   3506,   2881,   2401,   1921,   1296;
    65536,  48729,  40953,  35328,  30208,  24583,  16807;
  1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;
  ...

Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 6.
D. Foata and J. Riordan, Mappings of acyclic and parking functions, J. Aeq. Math., 10 (1974) 10-22.

Cf. A000272, A298593, A298594, A298597.

Table[Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

T(n,k) = Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).
T(n,k) = A298593(n,k)/n.
T(n,k) = Sum_{j=k..n} A298594(n,j).
T(n,k) = (Sum_{j=k..n} A298597(n,j))/n.
Sum_{k=1..n} T(n,k) = A000272(n+1).

A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 4, 6, 6, 27, 38, 36, 24, 256, 350, 330, 240, 120, 3125, 4202, 3960, 3000, 1800, 720, 46656, 62062, 58506, 45360, 29400, 15120, 5040, 823543, 1087214, 1025388, 806904, 546000, 312480, 141120, 40320, 16777216, 22024830, 20781690, 16524144, 11493720, 6985440, 3598560, 1451520, 362880

Offset: 1

Rui Duarte, Jan 22 2017

A parking function of size n is a sequence (a_1,...,a_n) of positive integers such that, if b_1 <= b_2 <= ... <= b_n is the increasing rearrangement of the sequence (a_1,..,a_n), then b_i <= i.
Given a:[n]->[n], the center of a is the largest subset Z(a) = { z_1, ..., z_k } of [n] such that z_1 < z_2 < ... < z_k and a_(z_j) <= j, for every j in [k]. The length of the center of a is |Z(a)|.
Then T(n,k)= number of parking functions of size n with center of length k.

			First seven rows:
      1
      1      2
      4      6      6
     27     38     36     24
    256    350    330    240    120
   3125   4202   3960   3000   1800    720
  46656  62062  58506  45360  29400  15120   5040

Rui Duarte and António Guedes de Oliveira, The number of parking functions with center of a given length, arXiv:1611.03707 (2016).

T(n,k) = k * A174551(n-1,k-1).
T(n,1) = (n-1)^(n-1) = A000312(n-1).
T(n,n-1) = n!(n-1)/2 = A001286(n), n>=2.
T(n,n) = n! = A000142(n).
Sum_{i=1,...,n} T(n,i) = (n+1)^(n-1) = A000272(n+1).

Table[Which[n == k == 1, 1, k == 1, (n - 1)^(n - 1), k == n, n!, True, k Sum[(-1)^j*Binomial[k - 1, j] (n - 1 - j)^(n - 1), {j, 0, k - 1}]], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 23 2017 *)

T(n,k) = k*Sum_{j=0..k-1} (-1)^j*binomial(k-1,j)*(n-1-j)^(n-1).

T(n,k) = k!*Sum_{j_1+j_2+...+j_k=n-k} (n-1)^(j_1)*(n-2)^(j_2)*...*(n-k)^(j_k).

A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 16, 0, 0, 5, 40, 16, 0, 0, 6, 80, 96, 0, 0, 0, 7, 140, 336, 64, 0, 0, 0, 8, 224, 896, 512, 0, 0, 0, 0, 9, 336, 2016, 2304, 256, 0, 0, 0, 0, 10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0, 11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Offset: 0

Rui Duarte and António Guedes de Oliveira, Sep 11 2013

Row n is the sum of the convolution of A089627(p,i) with A089627(n-p,i), for p=0,1,...,n.

			Triangle:
1
2, 0
3, 4, 0
4, 16, 0, 0
5, 40, 16, 0, 0
6, 80, 96, 0, 0, 0
7, 140, 336, 64, 0, 0, 0
8, 224, 896, 512, 0, 0, 0, 0
9, 336, 2016, 2304, 256, 0, 0, 0, 0
10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0
11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.

T(n,k) = 4^k * C(n+1, 2*k+1).
T(n,k) = sum(p=0..n, sum(i=0..k, C(p,i)*C(p-i, i)*C(n-p,k-i)*C(n-p-k+i, k-i))).
T(n,k) = A085841(n/2,k), if n is even.
T(n,k) = 2^k * A105070(n,k).

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User: Rui Duarte

A298593 Triangle read by rows: T(n,k) = number of times the value k appears on the parking functions of length n.

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A298594 Triangle read by rows: T(n,k) = number of parking functions a of length n such that a(1) = k and if we replace a(1) = k with k+1 we don't get a parking function.

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A298597 Number T(n,k) of times the value k appears on the parking functions of length n and such that if we replace that value k with k+1 we don't get a parking function.

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A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.

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Mathematica

Formula

A281485 Triangular array T(n,k) = k Sum_{j=0..k-1} (-1)^j binomial(k-1,j) (n-1-j)^(n-1), 1<=k<=n, read by rows.

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A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

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