A034841 -id:A034841 - cp's OEIS frontend

A331623 Number of sequences with n copies each of 1,2,...,n avoiding absolute differences between adjacent elements larger than one.

1, 1, 6, 92, 11482, 8956752, 54331653686, 2535604038015218, 951645881858020642746, 2911820015993491302722966990, 73784388170659542104264761249115686, 15642058800086197220958447712819197014917632, 27980772370697320617389378491983217784996780441605354

Offset: 0

Alois P. Heinz, Jan 22 2020

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 1.
a(2) = 6: 1122, 1212, 1221, 2112, 2121, 2211.
a(3) = 92: 111222333, 111223233, 111223323, 111223332, ..., 333221112, 333221121, 333221211, 333222111.

Alois P. Heinz, Table of n, a(n) for n = 0..25

Main diagonal of A331562.

Cf. A034841.

b:= proc(l, q) option remember; (n-> `if`(n<2, 1, add(
     `if`(l[j]=1, `if`(j in [1, n], b(subsop(j=[][], l),
     `if`(j=1, 0, n)), 0), b(subsop(j=l[j]-1, l), j)), j=
     `if`(q<0, 1..n, max(1, q-1)..min(n, q+1)))))(nops(l))
    end:
a:= n-> b([n$n], -1):
seq(a(n), n=0..6);

b[l_, q_] := b[l, q] = With[{n = Length[l]}, If[n < 2, 1, Sum[
      If[l[[j]] == 1, If[j == 1 || j == n, b[ReplacePart[l, j -> Nothing],
      If[j == 1, 0, n]], 0], b[ReplacePart[l, j -> l[[j]] - 1], j]], {j,
      If[q < 0, Range[n], Range[Max[1, q - 1], Min[n, q + 1]]]}]]];
a[n_] := b[Table[n, {n}], -1];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 8}] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)

a(n) = A331562(n,n).

a(9)-a(12) (using new data provided by Andrew Howroyd in A331562) from Alois P. Heinz, Sep 01 2020

A347810 Number of n-dimensional lattice walks from {n}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

Original entry on oeis.org

1, 1, 25, 2062017739, 255053951339165796439851848897794625

Offset: 0

Alois P. Heinz, Sep 14 2021

Lattice points may have negative coordinates, and different walks may differ in length. All walks are self-avoiding.

Alois P. Heinz, Table of n, a(n) for n = 0..6
Alois P. Heinz, Animation of a(2) = 25 walks
Wikipedia, Counting lattice paths
Wikipedia, Self-avoiding walk

Main diagonal of A347811.

Cf. A034841.

s:= proc(n) option remember;
     `if`(n=0, [[]], map(x-> seq([x[], i], i=-1..1), s(n-1)))
    end:
b:= proc(l) option remember; (n-> `if`(l=[0$n], 1, add((h-> `if`(
      add(i^2, i=h) b([n$n]):
seq(a(n), n=0..5);

s[n_] := s[n] = If[n == 0, {{}}, Sequence @@ Table[Append[#, i], {i, -1, 1}]& /@ s[n-1]];
b[l_List] := b[l] = With[{n = Length[l]}, If[l == Table[0, {n}], 1, Sum[With[{h = l+x}, If[h.h < l.l, b[Sort[h]], 0]], {x, s[n]}]]];
a[n_] := b[Table[n, {n}]];
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Nov 04 2021, after Alois P. Heinz *)

A378588 Triangle read by rows: T(n,k) is the number of maximal chains in the poset of all k-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 16, 22, 23, 1, 57, 94, 102, 103, 1, 226, 446, 507, 517, 518, 1, 961, 2308, 2764, 2855, 2867, 2868, 1, 4376, 12900, 16333, 17121, 17248, 17262, 17263, 1, 21041, 77092, 103666, 110487, 111739, 111908, 111924, 111925, 1, 106534, 489430, 701819, 761751, 773888, 775758, 775975, 775993, 775994, 1, 563961, 3282956, 5038344, 5578041, 5696293, 5716382, 5719046, 5719317, 5719337, 5719338

Offset: 1

John Tyler Rascoe, Dec 01 2024

			Triangle begins:
   k=1    2     3     4     5     6     7
 n=1 1;
 n=2 1,   2;
 n=3 1,   5,    6;
 n=4 1,  16,   22,   23;
 n=5 1,  57,   94,  102,  103;
 n=6 1, 226,  446,  507,  517,  518;
 n=7 1, 961, 2308, 2764, 2855, 2867, 2868;
 ...
T(3,3) = 6:
 () < (1) < (1,1) < (1,1,1),
 () < (1) < (1,1) < (1,2),
 () < (1) < (1,1) < (2,1),
 () < (1) < (2) < (1,2),
 () < (1) < (2) < (2,1),
 () < (1) < (2) < (3).

Cf. A034841, A143672, A282698, A317145, column k=2 A378382, main diagonal A378608.

def mchains(n,k):
    B,d1,S1 = [1,1],{(1,): 1},{(1,)}
    for i in range(n-1):
        d2,S2 = dict(),set()
        for j in S1:
            for x in [j+(1,), (1,)+j]+[j[:z]+tuple([j[z]+1])+j[z+1:] for z in range(len(j)) if j[z] < k]:
                if x not in S2: S2.add(x); d2[x] = d1[j]
                elif x != tuple([1]*(i+2)): d2[x] += d1[j]
        B.append(sum(d2.values())); d1 = d2; S1 = S2
    return B[:n+1]
def A378588_list(max_n):
    B = [mchains(max_n,i+1) for i in range(max_n)]
    return [[B[k][j+1] for k in range(j+1)] for j in range(max_n)]

T(n,k) = T(n,n) for k > n.

A378608 Number of maximal chains in the poset of all n-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 518, 2868, 17263, 111925, 775994, 5719338, 44592007, 366259499, 3157877470, 28492791496, 268307662047, 2630577754281, 26795670672626, 283038010150702, 3094882721541239, 34977231456293519, 407991690851302646, 4905431774834649852, 60721792897771836879

Offset: 0

John Tyler Rascoe, Dec 01 2024

nonn

			a(3) = 6:
  () < (1) < (1,1) < (1,1,1),
  () < (1) < (1,1) < (1,2),
  () < (1) < (1,1) < (2,1),
  () < (1) < (2) < (1,2),
  () < (1) < (2) < (2,1),
  () < (1) < (2) < (3).

Cf. A034841, A143672, A282698, A317145, A378382, main diagonal of A378588.

def mchains(n,k): return # See A378588
def A378608_list(max_n): return mchains(max_n,max_n)

A062716 Number of arrangements of 1,2,..,n*n in an n X n matrix such that each row is increasing or decreasing.

Original entry on oeis.org

1, 2, 24, 13440, 1009008000, 19947543780003840, 170891375144777551827763200, 942542805268120269309770939139883008000, 4650425326497486529782791149613966242353671284224000000

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001

nonn

Cf. A034841.

a(1) = 1 and for n >= 2: a(n) = 2^n * A034841(n) = 2^n * (n^2)! / (n!)^n.

More terms from Vladeta Jovovic, Jul 18 2001

A377135 Number of maximal chains in the poset of n-ary words of length n ordered by B covers A iff A_i <= B_i for 1 <= i <= n.

Original entry on oeis.org

1, 1, 2, 90, 369600, 305540235000, 88832646059788350720, 14007180988362844601443040716800, 1707750599894443404262670865631874246246400000, 217425846656446788579638892849417587480505167467321080630000000

Offset: 0

John Tyler Rascoe, Nov 26 2024

Terms are divisible by n for n > 0.

			For a(2) = (1,1) < (2,1) < (2,2), (1,1) < (1,2) < (2,2).
For n = 3 one chain is (1,1,1) < (1,2,1) < (1,2,2) < (1,2,3) < (1,3,3) < (2,3,3) < (3,3,3).

Cf. A000312, A034841, A300385, A378382.

a:= n-> (t-> (n*t)!/t!^n)(max(n-1, 0)):
seq(a(n), n=0..10);  # Alois P. Heinz, Nov 27 2024

a[n_]:=Product[Binomial[(n-1)*(n-i),n-1],{i,0,n-2}]; Array[a,10,0] (* Stefano Spezia, Nov 27 2024 *)

PARI
```
a(n) = {if(n<1,1,(n*(n-1))!/(n-1)!^n)}
```

a(n) = Product_{i=0..n-2} binomial((n-1)*(n-i),n-1).

a(n) = (n*(n-1))!/(n-1)!^n for n>=1, a(0)=1. - Alois P. Heinz, Nov 27 2024

A377601 Number of permutations of the multiset {1^n, 2^n,..., n^n} excluding permutations where all objects of all types are contiguous.

Original entry on oeis.org

0, 0, 4, 1674, 63062976, 623360743125000, 2670177736637149247308080, 7363615666157189603982585462030330960, 18165723931630806756964027928179555634194028453959680, 53130688706387569792052442448845648519471103327391407016237759999637120

Offset: 0

Mohammad Bakhshandeh, Nov 02 2024

			For n=2, the multiset is {1,1,2,2} and the a(2)=4 permutations counted are 1212, 1221, 2112, 2121 (but neither 1122 nor 2211).

Cf. A000142, A034841.

a(n) = (n^2)!/(n!)^n - n!.

a(n) = A034841(n) - A000142(n).

cp's OEIS Frontend

A331623 Number of sequences with n copies each of 1,2,...,n avoiding absolute differences between adjacent elements larger than one.

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A347810 Number of n-dimensional lattice walks from {n}^n to {0}^n using steps that decrease the Euclidean distance to the origin and that change each coordinate by at most 1.

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A378588 Triangle read by rows: T(n,k) is the number of maximal chains in the poset of all k-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

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A378608 Number of maximal chains in the poset of all n-ary words of length <= n, ordered by B covers A iff A_i <= B_{i+k} for all i in A and some k >= 0.

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A062716 Number of arrangements of 1,2,..,n*n in an n X n matrix such that each row is increasing or decreasing.

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Extensions

A377135 Number of maximal chains in the poset of n-ary words of length n ordered by B covers A iff A_i <= B_i for 1 <= i <= n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A377601 Number of permutations of the multiset {1^n, 2^n,..., n^n} excluding permutations where all objects of all types are contiguous.

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Examples

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