A030298 -id:A030298 - cp's OEIS frontend

A220656 The positions of those permutations in A030298 where the first element is not fixed.

3, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Correspondingly, gives the positions of those terms in A030299 whose first digit is not 1, as long as the decimal encoding system employed is valid.

Antti Karttunen, Table of n, a(n) for n = 1..5039

Complement: A220696. Cf. A081291.

(define (A220656 n) (+ (A003422 (+ 1 (A084558 n))) (A000142 (A084558 n)) (A212598 n)))

a(n) = A003422(1+A084558(n)) + A000142(A084558(n)) + A212598(n).

a(n) = A220655(n)+1.

A220696 The positions of those permutations in A030298 where the first element is one (fixed).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 12, 13, 14, 15, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Correspondingly gives the positions of those terms in A030299 whose first digit is 1, as long as the decimal encoding system employed is valid.

A. Karttunen, Table of n, a(n) for n = 1..5914

Complement: A220656. Cf. A072795.

a(1)=1; and for n>1, a(n)=A220695(n-1)+1.

A030488 Position of n-th 1 in A030298.

Original entry on oeis.org

1, 2, 5, 6, 9, 13, 17, 19, 23, 24, 28, 32, 36, 40, 44, 49, 53, 58, 63, 66, 71, 73, 77, 82, 87, 90, 95, 97, 101, 106, 111, 114, 119, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220

Offset: 1

Clark Kimberling

nonn

A030489 Position of n-th 2 in A030298.

Original entry on oeis.org

3, 4, 7, 11, 12, 15, 20, 22, 25, 29, 34, 39, 42, 47, 48, 52, 56, 60, 64, 68, 74, 79, 81, 85, 91, 94, 98, 103, 105, 109, 115, 118, 121, 126, 131, 136, 141, 146, 152, 157, 163, 169, 173, 179, 182, 187, 193, 199, 203, 209, 212, 217, 223

Offset: 1

Clark Kimberling

nonn

A030490 Position of n-th 3 in A030298.

Original entry on oeis.org

8, 10, 14, 16, 18, 21, 26, 31, 33, 37, 43, 46, 50, 55, 57, 61, 67, 70, 72, 76, 80, 84, 88, 92, 99, 102, 107, 110, 113, 117, 122, 127, 133, 139, 143, 149, 151, 156, 161, 166, 171, 176, 183, 189, 192, 197, 204, 208, 213, 219, 222, 227

Offset: 1

Clark Kimberling

nonn

Cf. A030298.

A030491 Position of n-th 4 in A030298.

Original entry on oeis.org

27, 30, 35, 38, 41, 45, 51, 54, 59, 62, 65, 69, 75, 78, 83, 86, 89, 93, 96, 100, 104, 108, 112, 116, 123, 129, 132, 137, 144, 148, 153, 159, 162, 167, 174, 178, 181, 186, 191, 196, 201, 206, 214, 218, 224, 228, 232, 237, 243, 249

Offset: 1

Clark Kimberling

nonn

A030492 Position of n-th 5 in A030298.

Original entry on oeis.org

124, 128, 134, 138, 142, 147, 154, 158, 164, 168, 172, 177, 184, 188, 194, 198, 202, 207, 211, 216, 221, 226, 231, 236, 244, 248, 254, 258, 262, 267, 274, 278, 284, 288, 292, 297, 304, 308, 314, 318, 322, 327, 331, 336, 341

Offset: 1

Clark Kimberling

nonn

A030493 n-th partial sum of A030298.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 13, 14, 17, 19, 21, 22, 25, 27, 30, 31, 34, 35, 37, 40, 42, 43, 44, 46, 49, 53, 54, 56, 60, 63, 64, 67, 69, 73, 74, 77, 81, 83, 84, 88, 90, 93, 94, 98, 101, 103, 105, 106, 109, 113, 115, 116, 120, 123, 125, 128, 129

Offset: 1

Clark Kimberling

nonn

A347208 Number of permutations of [n] that are in the same position in A030298 as they are in A098281.

Original entry on oeis.org

1, 2, 3, 3, 3, 10, 5, 4, 5, 13, 3, 6, 5

Offset: 1

William Chang, Aug 23 2021

			For n = 7 the a(7) = 5 permutations are 1234567, 1234576, 1236745, 6574231, 7654321, which are in positions 1, 2, 17, 4198, 5040 respectively in both sequences A030298 and A098281.

Cf. A030298, A098281.

perms[n_] := perms[n] = If[n == 1, {{1}}, Flatten[Table[Insert[#, n, pos], {pos, -1, -n, -1}]& /@ perms[n-1], 1]];
a[n_] := Count[Transpose@{perms[n], Permutations[Range[n]]}, {p_, p_}];
Table[a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 02 2021 *)

A356324 a(n) is the first split point of the permutation p if p is the n-th permutation (in lexicographic order (A030298 prepended by the empty permutation)), or zero if it has no split point.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 4, 0, 0

Offset: 0

Peter Luschny, Aug 03 2022

A permutation p in [n] (where n >= 0) is reducible if there exist an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation p at i and that i is a split point of p.
The list of permutations starts with the empty permutation (), which has no split points. The first permutation which has a split point is (1, 2).
The number of terms corresponding to the permutations of [n] which vanish is A003319(n), and the numbers of nonzero terms is A356291(n).

			Rows give the terms corresponding to the permutations of [n].
[0] [0]
[1] [0]
[2] [1, 0]
[3] [1, 1, 2, 0, 0, 0]
[4] [1, 1, 1, 1, 1, 1, 2, 2, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Cf. A003319, A356291, A059438, A030298.

def FirstSplit(p) -> int:
    n = p.size()
    for i in (1..n-1):
        ok = True
        for j in (1..i):
            if not ok: break
            for k in (i + 1..n):
                if p(j) > p(k):
                    ok = False
                    break
        if ok: return i
    return 0
def A356324_row(n): return [FirstSplit(p) for p in Permutations(n)]
for n in range(6): print(A356324_row(n))

cp's OEIS Frontend

A220656 The positions of those permutations in A030298 where the first element is not fixed.

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A220696 The positions of those permutations in A030298 where the first element is one (fixed).

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A030488 Position of n-th 1 in A030298.

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A030489 Position of n-th 2 in A030298.

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A030490 Position of n-th 3 in A030298.

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A030491 Position of n-th 4 in A030298.

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A030492 Position of n-th 5 in A030298.

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A030493 n-th partial sum of A030298.

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A347208 Number of permutations of [n] that are in the same position in A030298 as they are in A098281.

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A356324 a(n) is the first split point of the permutation p if p is the n-th permutation (in lexicographic order (A030298 prepended by the empty permutation)), or zero if it has no split point.

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