A000680 -id:A000680 - cp's OEIS frontend

A327410 Numbers represented by the partition coefficients of prime partitions.

1, 6, 10, 20, 21, 36, 56, 78, 90, 105, 120, 171, 210, 252, 300, 364, 465, 528, 560, 741, 756, 792, 903, 990, 1140, 1176, 1485, 1540, 1680, 1830, 1953, 1980, 2346, 2520, 2600, 2628, 2775, 3240, 3432, 3570, 4095, 4368, 4851, 4960, 5253, 5460, 5886, 5984, 6105

Offset: 1

Peter Luschny, Sep 07 2019

nonn

Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by . We say 'k is represented by pi' if k = .

A partition is a prime partition if all parts are prime.

			(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1   <- [2],
6   <- [2, 2],
10  <- [3, 2],
20  <- [3, 3],
21  <- [5, 2],
36  <- [7, 2],
56  <- [5, 3],
78  <- [11, 2],
90  <- [2, 2, 2],
105 <- [13, 2],
120 <- [7, 3],
171 <- [17, 2],
210 <- [3, 2, 2],
252 <- [5, 5],
300 <- [23, 2].

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition

Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.

def A327410_list(n):
    res = []
    for k in range(2*n):
        P = Partitions(k, parts_in = prime_range(k+1))
        res += [multinomial(p) for p in P]
    return sorted(Set(res))[:n]
print(A327410_list(20))

A358112 Table read by rows. A statistic of permutations of the multiset {1,1,2,2,...,n,n}.

Original entry on oeis.org

1, 5, 1, 47, 42, 1, 641, 1659, 219, 1, 11389, 72572, 28470, 968, 1, 248749, 3610485, 3263402, 357746, 4017, 1, 6439075, 204023334, 371188155, 95559940, 3853617, 16278, 1, 192621953, 12989570167, 43844432805, 22448025251, 2216662051, 38270373, 65399, 1

Offset: 1

Peter Luschny, Oct 30 2022

Table 1, page 12 in Maazouz and Pitman (note a typo in T(2, 0)).

			[n\d]    0            1           2           3           4           5     6
-----------------------------------------------------------------------------
[1]         1;
[2]         5,           1;
[3]        47,          42,           1;
[4]       641,        1659,         219,           1;
[5]     11389,       72572,       28470,         968,          1;
[6]    248749,     3610485,     3263402,      357746,       4017,        1;
[7]   6439075,   204023334,   371188155,    95559940,    3853617,    16278, 1
[8] 192621953, 12989570167, 43844432805, 22448025251, 2216662051, 38270373,
65399, 1

Yassine El Maazouz and Jim Pitman, The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution, arXiv:2210.02027 [math.PR], Oct. 2022.

Cf. A006902 (row 0), A000680 (row sums).

P := (n, x) -> (2*n)!*add(add(binomial(n, k)*binomial(n-k, j)*
(-1)^(n-k-j)*max(x - k, 0)^(2*n - j)/(2*n - j)!, j = 0..n-k), k = 0..n):
Trow := n -> seq(P(n, k+1) - P(n, k), k = 0..n-1):
seq(print(Trow(n)), n = 1..8);

T(n, k) = P(n, k+1) - P(n, k), where P(n, x) = (2*n)!*Sum_{k=0..n} Sum_{j=0..n-k} binomial(n, k)*binomial(n-k, j)*(-1)^(n-k-j)*max(x - k, 0)^(2*n - j)/(2*n - j)!.

A368028 Square array read by antidiagonals; T(n,k) = number of ways a vehicle with capacity k can transport n distinct individuals with distinct starting and finishing points.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 6, 6, 1, 1, 0, 24, 54, 6, 1, 1, 0, 120, 648, 90, 6, 1, 1, 0, 720, 9720, 1944, 90, 6, 1, 1, 0, 5040, 174960, 52920, 2520, 90, 6, 1, 1, 0, 40320, 3674160, 1730160, 99000, 2520, 90, 6, 1, 1, 0, 362880, 88179840, 65998800, 4806000, 113400, 2520, 90, 6, 1, 1, 0, 3628800, 2380855680, 2877275520, 274050000, 6966000, 113400, 2520, 90, 6, 1, 1

Offset: 0

Henry Bottomley, Dec 24 2023

			T(3,2)=54 represented by the nine patterns AABBCC, AABCBC, AABCCB, ABABCC, ABACBC, ABACCB, ABBACC, ABBCAC, ABBCCA multiplied by 3!=6 for the permutations of A,B,C; but for example ABCABC would not work as the vehicle would be over its capacity of 2 after picking up 3 passengers.

math.stackexchange, Passenger entrance/exit combinations

Cf. A080934. Rows include A000012, A057427. Columns include A000007, A000142, A034001. Diagonals include A000680 and A071798.

If f(n,k,c)=n*f(n-1,k,c+1)+c*f(n,k,c-1) with f(n,k,c)=0 when n<0 or k<0 or c<0 or k

A375220 T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

Original entry on oeis.org

5, 74, 15, 2193, 296, 30, 101644, 10965, 740, 50, 6840085, 609864, 32895, 1480, 75, 630985830, 47880595, 2134524, 76755, 2590, 105, 76484389121, 5047886640, 191522380, 5692064, 153510, 4144, 140, 11792973495032, 688359502089, 22715489880, 574567140, 12807144, 276318, 6216, 180

Offset: 2

Hugo Pfoertner, Aug 08 2024

			The triangle begins
          5,
         74,       15,
       2193,      296,      30,
     101644,    10965,     740,    50,
    6840085,   609864,   32895,  1480,   75,
  630985830, 47880595, 2134524, 76755, 2590, 105

Cf. A000217, A000680, A028895, A116218, A374980 (column 0), A375222 (column 1), A375223.

Cf. A375219 (similar for triples in the multiset).

\\ using functions mima and a375219 from A375219, row n of triangle:
a375219(n,sizeb=2)

T(n,n) = 1, T(n,n-1) = 0 (terms not in DATA),
T(n,n-2) = 5*n*(n-1)/2 = 5*A000217(n-1) = A028895(n-1),
Sum_{j=0..n-2} T(n,j) = (2*n)!/(2^n) - 1 = A000680(n) - 1,
Sum_{j=1..n-2} T(n,j) = A375223(n) - 1.

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A327410 Numbers represented by the partition coefficients of prime partitions.

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A358112 Table read by rows. A statistic of permutations of the multiset {1,1,2,2,...,n,n}.

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A368028 Square array read by antidiagonals; T(n,k) = number of ways a vehicle with capacity k can transport n distinct individuals with distinct starting and finishing points.

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A375220 T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

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