A226873 -id:A226873 - cp's OEIS frontend

A292720 Number of multisets of nonempty words with a total of n letters over 5-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

1, 1, 4, 14, 67, 343, 1431, 6620, 31539, 151680, 769374, 3586756, 17500630, 85727012, 420986605, 2116435479, 10254063794, 50697425138, 251055167912, 1244053731675, 6246442090103, 30737278735067, 152890117563022, 761050222982081, 3790169351183134

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A292712.

Cf. A226875, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 5), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226875(j).

Euler transform of A226875.

A292721 Number of multisets of nonempty words with a total of n letters over 6-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 9860, 53739, 279360, 1595454, 8733436, 53035750, 280946972, 1626421033, 9103196607, 53266673126, 300953629850, 1817236258604, 10114067087883, 59666856128423, 342703064143223, 2024687556279346, 11644875879288821, 70172335165701018

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A292712.

Cf. A226876, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 6), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226876(j).

Euler transform of A226876.

A292722 Number of multisets of nonempty words with a total of n letters over 7-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 78939, 470880, 2805054, 17296396, 110869750, 716116412, 4868928433, 30169477167, 196213489166, 1272224938490, 8371053970724, 55530391931787, 369148740629927, 2514270810574079, 16452249276575722, 109991261928916853

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=7 of A292712.

Cf. A226877, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 7), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226877(j).

Euler transform of A226877.

A292723 Number of multisets of nonempty words with a total of n letters over 8-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 692640, 4659774, 30077836, 209311030, 1433872892, 10689029713, 76772260527, 600293120366, 4142024767610, 30775147154084, 221566161988587, 1663421685691847, 12221938274124959, 93706886872251562, 696726353909296853

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A292712.

Cf. A226878, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 8), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226878(j).

Euler transform of A226878.

A292724 Number of multisets of nonempty words with a total of n letters over 9-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 1055520, 6837054, 49975756, 358031350, 2673108092, 20399335633, 161247005007, 1321885836686, 10814140769210, 93349395210404, 726371063425227, 5939975798740967, 48195816632614079, 396235068140514442

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=9 of A292712.

Cf. A226879, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 9), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226879(j).

Euler transform of A226879.

A292725 Number of multisets of nonempty words with a total of n letters over 10-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 4, 14, 67, 343, 2151, 14900, 119259, 1055520, 10465854, 73562956, 592088950, 4560084092, 37322365393, 303133205967, 2640424710926, 22786686453050, 210764523790244, 1891228958070987, 18197644702881767, 155143878113188799, 1411297482751989322

Offset: 0

Alois P. Heinz, Sep 21 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=10 of A292712.

Cf. A226880, A226873.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*d!*
      b(d, 0, 10), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

G.f.: Product_{j>=1} 1/(1-x^j)^A226880(j).

Euler transform of A226880.

A381349 Triangle read by rows: T(n,k) is the number of distinct tuples E each corresponding to some k-ary word W = (w_1, ..., w_n), where E is a tuple (e_1, ..., e_{n-1}) with e_i being the number of pairs of equal letters (w_j,w_k) in W such that j + i = k.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 6, 9, 10, 1, 10, 22, 26, 27, 1, 20, 54, 73, 78, 79, 1, 36, 163, 249, 269, 275, 276, 1, 72, 447, 791, 915, 942, 949, 950, 1, 135, 1350, 3136, 3776, 3899, 3934, 3942, 3943, 1, 272, 4088, 11315, 14849, 15650, 15811, 15855, 15864, 15865

Offset: 1

John Tyler Rascoe, Feb 21 2025

Two different words of the same length can have the same corresponding E tuple.

			Triangle begins:
    k=1   2    3    4    5    6    7
  n=1 1;
  n=2 1,  2;
  n=3 1,  3,   4;
  n=4 1,  6,   9,  10;
  n=5 1, 10,  22,  26,  27;
  n=6 1, 20,  54,  73,  78,  79;
  n=7 1, 36, 163, 249, 269, 275, 276;
  ...
Considering the pairs of equal letters in W = (1,3,3,1) there is 1 pair with no letters between them, 0 pairs with a single letter between them, and 1 pair seperated by two letters, giving E = (1,0,1).
T(4,2) = 6 counts the following values of E each listed with a corresponding word:
     E        W
  (3,2,1) (1,1,1,1)
  (2,1,0) (1,1,1,2)
  (2,0,0) (1,1,2,2)
  (1,0,1) (1,2,2,1)
  (1,1,1) (2,1,2,2)
  (0,2,0) (2,1,2,1)

Michael S. Branicky, Table of n, a(n) for n = 1..78 (terms through row 12)
Michael S. Branicky, Python Program

Cf. A000312, (empirical column k=2) A006606, A120910, A226873.

from itertools import combinations_with_replacement, permutations
def pairs(m): return tuple([sum(1 for j in range(len(m)-i) if m[j] == m[j+i]) for i in range(1,len(m))])
def A381349(n,k):
    S = set()
    for y in combinations_with_replacement(range(1,k+1),n-1):
        S.update(z for z in permutations((1,)+y))
    return len({pairs(i) for i in S})

# see links for a different algorithm
from sympy.utilities.iterables import multiset_permutations
from itertools import combinations, combinations_with_replacement
def E(w):
    return tuple(sum(1 for j, k in combinations(range(len(w)), 2) if w[j] == w[k] and j+i == k) for i in range(len(w)))
def row(n): # generator of row n
    v, S = 0, set()
    for k in range(1, n+1):
        for c in combinations_with_replacement(range(1, k+1), n-k):
            if len(c) > 0 and c[0] == 2: break
            S.update(E(e) for e in multiset_permutations(tuple(range(1, k+1))+c))
        yield len(S)
print([e for n in range(1, 9) for e in row(n)]) # Michael S. Branicky, Feb 26 2025

T(n,k) = T(n,n) for k > n.
From Michael S. Branicky, Feb 25 2025: (Start)
T(n,   2) = A006606(n).
T(n, n-1) = T(n, n-2) + n-1, arising from E's of permutations of 1, ..., n-2, n-1, n-1.
T(n,   n) = T(n, n-1) + 1, arising from E(1, ... , n) = (0, ..., 0). (End)

a(37) and beyond from Michael S. Branicky, Feb 25 2025

cp's OEIS Frontend

A292720 Number of multisets of nonempty words with a total of n letters over 5-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A292721 Number of multisets of nonempty words with a total of n letters over 6-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A292722 Number of multisets of nonempty words with a total of n letters over 7-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A292723 Number of multisets of nonempty words with a total of n letters over 8-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A292724 Number of multisets of nonempty words with a total of n letters over 9-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A292725 Number of multisets of nonempty words with a total of n letters over 10-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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Maple

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A381349 Triangle read by rows: T(n,k) is the number of distinct tuples E each corresponding to some k-ary word W = (w_1, ..., w_n), where E is a tuple (e_1, ..., e_{n-1}) with e_i being the number of pairs of equal letters (w_j,w_k) in W such that j + i = k.

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